Some time ago I surmised that it might be possible to reactivate the
adhesive between the two plastic layers by ironing with an iron set
on polyester.

Well, you can, at least if it hasn't gone too far.  Takes care and
thought, works best on the blue side (it's actually the adhesive
that's blue, it would seem) which is much thinner, and you may need
to hold things together for a few seconds after you take the iron
away.  A thumb works -- it's hot but doesn't have enough thermal mass
to actually burn you.

I doubt (though I'm not sure) this would work to reattach the ends of
the foils where they're exposed at the connector.

Yours,
David