Can they enquire? Most likely yes.
Can they require that information? Not directly. If it's a condition of taking care, they can refuse to take over (but I think this is unlikely).
More likely than not I would think they're simply making sure everything is above board and, if not, will look to refer the matter to the Office of the Public Guardian. I would think the Public Trustee has a remit to look out for instances of financial abuse, and a situation where there is not a formal power of attorney in place is a warning sign.