I honestly can't fathom why Firefox's cache isn't doing what it's supposed to.

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I looked into "Service Workers" and tried a few things here and there.
For the most part, they're.. Umm..
Well, this guy put it best.
"A sledgehammer to crack a nut"..

Indeed..

So, instead, I'm just going to carry on using AppCache, and ignore Firefox's issue.
From what I can tell, it isn't a case of the cache not being available. The files are definitely being cached. I can see them using a "inspect the cache" tool that I found online!!
Instead, it's that Firefox isn't then reading the cache when it needs to access them again, and that's got me baffled.
So, for now, I'm going to ignore it.

I'll probably stick a "Sorry, this is broken in Firefox, please use ANY other browser, even Edge!" notice on the page.

Hate to do that, but I'm honestly struggling with the thing, and it's taken up FAR too long to figure it all out.