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Drill the grub screw out?
Can't access the screw on the soap dish with a drill. Could just hacksaw the case to bits I guess; it's dead anyway.
The towel holder actually spins on a central axis, it's as if the screw is doing nothing. Have you got any other photos of the bracket? I can't tell how it works.
There's three of these: the cup / TB holder, towel holder and soap holder and the screw on all of them turns but does not release. How is that possible?!
Howard-
Unscrew the grub screw completely, it might still be holding the bracket even though you have unscrewed it a bit. There's not much to it, if that grub screw comes out then you should be able to remove the accessory before removing the plate behind.
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There'll be something like this attached to the wall that the grub screw will cinch up to
Airhead
aggi
There'll be a bracket that it tightens onto that's screwed into the wall. Grub screw will be like so:
Drill the screw out and it'll come out