if you are in same case as me.
normally if you open your WebPage TWICE (the page where there is the Iframe) on your web browser,
you should discover with surprize that Qlik conserving your selection ;-)
Of course that's not a Solution !
but if that the case, then the thing to do is "easy" on your HTML page make not One, but 2 iframe (of course the 2nd iframe must be hidden by using any way : reduce size to 1 px, or using CSS with display : none...)
It should fix your issue