Thanks RikuP, those were good and promising suggestions. But sadly they didn't fix it for me (on both Chrome and Opera). It even pauses when the game goes behind another window without minimizing it. The only thing that seems to work is to keep it visible.
The Recruit Unit button does nothing and causes the following error to appear in the browser's developer console:
ERROR core/src/avm2/events.rs:414 Error dispatching event EventObject { type: "click", class: flash.events::MouseEvent, ptr: 0x192df8a0 } to handler FunctionObject { ptr: 0x170f7db8, name: "_-5z::_-7p/_-W()" } : TypeError: Error #1010: A term is undefined and has no properties. (accessing field: stats) at com::_-BY/KongUpdateStats() at _-HH::_-Tk/NumericAchievementCheck() at _-5z::_-4P/confirmRecruit() at _-5z::_-7p/_-W()
It's trying to access the 'stats' property of an uninitialized object.
That may or may not be caused by this error immediately before:
Failed to load resource: net::ERR_FAILED
ruffle_web.js:2520 ERROR core/src/loader.rs:1683 Error during URLLoader load of "https://com-lvlboost-incoming.herokuapp.com/v1?publickey=ebbb1485": FetchError("Got JS error")
Try to open that URL in a browser and it says "No such app. There's nothing here yet..."
 "GrindQuest Censored"){kind=icon}
 "The King's League: Odyssey"){kind=icon}