If Caroline is shielding and has to work at Jack’s home then that is a valid furlough claim - i has missed that. If she was a PA that could do something useful, like social media management, which could be done remotely, that’s where I think it gets iffy. But then if she has childcare issues again valid furlough claim. Employers can pay 100% wages and will only get 80% back, actually less as time goes on so she could be out of pocket a bit. If she is claiming the company must be eligible.
She has been working throughout so her claim to be #unsupported doesn’t really wash.