I solved the problem by setting a z-index on the header. It seems that just having the z-index of the menus higher than the page content was not sufficient.
BTW, I see a z-index of 10000 and 10001 in /slideshow-jquery-image-gallery/style/SlideshowPluginSlideInserter/slide-inserter.css. I didn’t parse your code far enough to see what elements that was styling. I just shot for something higher.
Thanks for having a look at the issue. It is working now.