The sidebar tabs are defined in $:/core/ui/SideBarSegments/tabs and are produced using the tags macro.
In this macro, if the retain="yes" parameter is specified, then “the content of the tabs should be retained when switching to another tab, avoiding re-rendering it (this can be useful to avoid video or audio sources unexpectedlyresetting)” (see https://tiddlywiki.com/#tabs%20Macro)
Thus, if you add retain="yes" to the $macrocall in $:/core/ui/SideBarSegments/tabs, it should also retain the rendered IFRAMEs for your TiddlyTools MiniBrowser tabs.
Note that retaining the sidebar tabs may slow down your TiddlyWiki, since it will also retain the “recent” tab, which is refreshed every time a tiddler change occurs.
-e