Your gallery's height is currently expressed as a percentage.
This will be a percentage of the gallery's parent container but, when the page is first loaded, the gallery's parent container is not visible and Juicebox may see its height as being 0px. 60% x 0px = 0px which may explain why the gallery is not displayed.

Instead, try expressing your gallery's height as a fixed pixel value (e.g. 600px) to see if this makes a difference.

If this does not help, then try also adding the gallery's height as inline CSS to the gallery container div, ie.:

<div id="juicebox-container" style="height: 600px;"></div>

... similar to the 'Using Percentage Heights' suggestion here.

Also, as you have multiple galleries which may be displayed on the same page, make sure you embed each gallery into a container with a unique id.

For example:

<script type="text/javascript" src="http://www.monpetitblog.net/galerie/2013_11_juice/jbcore/juicebox.js">

<script type="text/javascript">
new juicebox({
baseUrl : 'http://www.monpetitblog.net/galerie/path_to_gallery/',
containerId : "juicebox-container1",
galleryTitle: "Sept.'13",
galleryWidth: "100%",
galleryHeight: "600",
});
</script>
<div id="juicebox-container1" style="height: 600px;"></div>

<script type="text/javascript">
new juicebox({
baseUrl : 'http://www.monpetitblog.net/galerie/path_to_another_gallery/',
containerId : "juicebox-container2",
galleryTitle: "Sept.'13",
galleryWidth: "100%",
galleryHeight: "600",
});
</script>
<div id="juicebox-container2" style="height: 600px;"></div>

Incidentally, as you have WP-Juicebox activated, the 'juicebox.js' JavaScript file will be automatically loaded on your pages so there is no need to include the file in each and every gallery you manually embed.
(If you were to deactivate WP-Juicebox, you would need to include the file in each gallery's embedding code.)

If you were using Juicebox-Pro, you could set expandInNewPage="TRUE" which would force the gallery to expand in a new page (instead of expanding it in the page into which it is embedded). This should avoid the problem of returning to the same point in your infinite scroll page. Please see here for more details.
Unfortunately, expandInNewPage is a Pro option only and is not available to Juicebox-Lite (the free version).

This is because, in Juicebox-Lite, expandInNewPage is set to AUTO and on iOS devices, expanded galleries are opened in a new page by default. (Please see the link above for details.)

In order to solve the issues you are experiencing, perhaps you will need to delay the loading of the Juicebox galleries on the page until the gallery's parent containers are visible.
As you previously mentioned:

If you could somehow hook into the Infinite Scroll plugin and run a custom JavaScript function when the appropriate containers are displayed on screen, you could either reassign dimensions to the gallery (using JavaScript and CSS) or use the Juicebox-Pro Pro API method showGallery() to try to force the gallery to be displayed. Please note that the Juicebox-Pro Pro API is available to Juicebox-Pro only (and not to Juicebox-Lite).

Just for the record, I do not see this when viewing your web page for myself. Expanding the September gallery displays the September gallery (not the November gallery) in both Chrome and Firefox.

Juicebox itself does not fire a page refresh when a gallery is loaded so I am not sure what might be causing this.
Does this happen in Chrome on your iDevice(s) or just in Mobile Safari?
I notice that the code on your webpage does not validate correctly. Perhaps fixing the HTML errors on your web page will make a difference.
Try running your web page's URL through the W3C Markup Validation Service and fix the errors reported.
Even if this does not help, we can then eliminate HTML errors as a possible cause of the issues you are experiencing.

The only oddity I see on your web page at the moment is that your embedding code (for your November gallery) is enclosed in CDATA tags and that the closing tag has its final > character escaped as > in the following line:

// ]]&gt;</script></p>

It may (or may not) make a difference but try either replacing the line above with:

// ]]>
</script>
</p>

... or completely remove the CDATA tags around the JavaScript code.

Is it OK in both Chrome and Mobile Safari on the iPad? What iOS does your iPad run? Perhaps you can upgrade the version of iOS on your iPhone (if it is not already up-to-date) to see if this makes a difference. (By checking both Chrome and Mobile Safari in both an iPhone and an iPad and checking which version of iOS each device is running, you may at least be able to find a pattern within iOS devices.)

As you seem to be embedding your galleries manually (rather than using WP-Juicebox), then perhaps a workaround would be to upload your gallery folders to your web server and include links to them within your posts (rather than embed them directly in your posts).

I have been able to replicate the problem you describe and it seems to be directly related to the use of the Splash Page.

When a Juicebox gallery is displayed on a mobile device, Small Screen Mode is used and, by default, the Splash Page is displayed. This may not happen on an iPad with a larger screen (where Large Screen Mode may be used) which is perhaps why you do not see the problem on your iPad.
(For more information about Screen Modes, please see here.)

If I set showSplashPage="NEVER" (to prevent the Splash Page from being displayed) or screenMode="LARGE" (to display the Large Screen Mode version of the gallery which, by default, does not display the Splash Page), the infinite scroll page does not jump back up to the top.

I have contacted the developers regarding this issue and hopefully it can be fixed in a future version of Juicebox but in the meantime, a workaround would be to prevent the Splash Page from being displayed (by setting either showSplashPage="NEVER" or screenMode="LARGE").
Please note that both of these configuration options are available in Juicebox-Pro only and are not supported in Juicebox-Lite (the free version).

I am a Pro user now! ;-)

So I am able to create November (page 1), October (page 2) and September (page 3) galleries using expandInNewPage="TRUE" and showSplashPage="NEVER" options. Refresh does not happen anymore but I have some concerns about expandInNewPage...

• What I can see is that the option does not allow us to open a new tab but the gallery will stay on the same page in some way. As a consequence, I still see a weird display of my September gallery post (main image is very little, with no thumbnails) after I opened November gallery first (from the corresponding post). Note that October post has a good display... To make it even more complex, this issue happens within Chrome and Safari but everything seems OK with Firefox (all are iMac version).

• I notice that when I go to my main page, the http://www.monpetitblog.net web address is replaced by http://www.monpetitblog.net/#1. Do you have any explanation? Could it be related to the issue explained above?

• On iPad, after closing a gallery, I am often in a situation in which the infinite scroll won't work anymore within the page...

• Is there any option to open the gallery in a new tab? Then closing the gallery would mean closing the tab, and the main website would appear again after that. I am sure it would make my issues disappear...

Thank you in advance.

Regards.