Rather than setting autoPlayOnLoad="TRUE", use autoPlayOnLoad="FALSE" (the default value) and start the AutoPlay via the Juicebox-Pro API toggleAutoPlay() method when the onInitComplete event is fired (once the gallery is ready).

At the moment, you'll need to also set enableAutoPlay="TRUE" for this to work. This should not be necessary: the toggleAutoPlay() API method should not have the enableAutoPlay configuration option as a dependency. However, the developers are aware of this issue and it should hopefully be fixed in the next version.

Also, be sure to set goNextOnAutoPlay="FALSE" (default value), otherwise, when AutoPlay is started, the image will change immediately instead of waiting the allotted displayTime before moving on.

Try the following:

<!--START JUICEBOX EMBED-->
<script src="jbcore/juicebox.js"></script>
<script>
    var jb = new juicebox({
        containerId: "juicebox-container",
        autoPlayOnLoad: "FALSE",
        enableAutoPlay: "TRUE",
        goNextOnAutoPlay: "FALSE"
    });
    jb.onInitComplete = function() {
        jb.toggleAutoPlay();
    };
</script>
<div id="juicebox-container"></div>
<!--END JUICEBOX EMBED-->

Hopefully this will help.
If not, then you could introduce a short delay using the JavaScript setTimeout() function.

For example:

<!--START JUICEBOX EMBED-->
<script src="jbcore/juicebox.js"></script>
<script>
    var jb = new juicebox({
        containerId: "juicebox-container",
        autoPlayOnLoad: "FALSE",
        enableAutoPlay: "TRUE",
        goNextOnAutoPlay: "FALSE"
    });
    jb.onInitComplete = function() {
        window.setTimeout(function() {
            jb.toggleAutoPlay();
        }, 500);
    };
</script>
<div id="juicebox-container"></div>
<!--END JUICEBOX EMBED-->

The '500' entry in the code above is the delay in milliseconds. You can change this to a different value if you like.

The only other thing I can think of would be to start AutoPlay when the Juicebox-Pro API determines that the first image has been displayed for the first time (using the onImageChange event).
Try the following:

<!--START JUICEBOX EMBED-->
<script src="jbcore/juicebox.js"></script>
<script>
    var jb = new juicebox({
        containerId: "juicebox-container",
        autoPlayOnLoad: "FALSE",
        enableAutoPlay: "TRUE",
        goNextOnAutoPlay: "FALSE"
    });
    var flag = false;
    jb.onImageChange = function(e) {
        if (flag === false && e.id === 1) {
            jb.toggleAutoPlay();
            flag = true;
        }
    };
</script>
<div id="juicebox-container"></div>
<!--END JUICEBOX EMBED-->

If this still does not help, then you might need to use a setTimeout() delay (as I noted in my post above) or tweak your gallery's imagePreloading value and/or your image sizes (although I understand that you do not want to do this).

You can set configuration options in the gallery's XML file, in the embedding code or via a query string in the URL.
Please see the Setting Config Options support section for details.
If you have a certain configuration option set in both the embedding code and the XML file, then the value from the embedding code will take precedence. (Options set via the query string will override both embedding code and XML file options.)
I chose to add the configuration options required for my code to function correctly to the gallery's embedding code simply to keep my suggestion to a single chunk of code. You can set all your configuration options in your gallery's 'config.xml' file without any problem.

It looks like this might be the best course of action (other than changing the filesize of your images).

As a side note, you might like to look into having a Multi-Size Image gallery. Please see the Multi-Size Image Support section for details.
In a Multi-Size Image gallery, up to 3 different sizes of images can be used and Juicebox-Pro chooses the most appropriate image size depending on factors such as the device, screen size and pixel density being used to view the gallery. The speed of the connection is not a factor in deciding which image size to use but at least you should be able to feed smaller images to mobile devices to minimize the problem as much as you can.

OK. No problem.