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I'm no expert on this but my instinct would be to drill a small exploratory hole to try to find something solid behind it, then once I know the depth of the good stuff, fix some kind of threaded fitting in to that that the rail can be attached to, then patch the rest of it up with filler, then repaint. The entire room. Le sigh.
I guess it's clear that the plaster can't take it, presumably because of the movement and stress that happens when the curtains are pulled / unpulled, so you need to find something more solid to get in to.
Or just patch it and fit blinds or, heaven forbid, shutters.
Not directly, I think there’s a gap and it’s probably Victorian lathe with a couple of different skim coats on top.