despite research and chats my dad is still keen to keep her at home.
This is the first problem. As long as your dad says he will and can care for her at home (and you and you sisters are propping him up), ss will go along with that. It’s your dad that has to decide this situation is no longer tenable for him and for her and state this out loud.
Does your mum have capacity to make a decision about where she lives ? If so, then she gets to choose - but you and your dad also get to choose whether you support it. Withdraw care, tell ss there has been total carer breakdown, and they have to step in. They ultimately have a duty of care to her that you do not. You and your dad can walk away, rescind POA, leave them to it and you are not legally in the wrong.
If your mum has lost capacity…
You have POA - does your dad? If he doesn’t and it’s only you, then in theory you can overrule him, especially if you believe that staying at home is not in your mums best interests. Do you have control of her finances, and an understanding of her assets?
What did the care assessment recommend? Four care visits a day, to be funded by your mum and arranged by you, I’m guessing. But it’s not enough to help your dad have any quality of life.
The bottom line is that as long as you and your dad are keeping her ‘safe’ at home, ss are under no pressure. Being dirty is not the same as unsafe, though it’s obviously horrible for your dad. Unsafe is wandering, turning the gas on, having falls and accidents. People live alone in horrible, dirty conditions all the time but if they are choosing to do so, and if family are present to deal with it, ss are not obliged to intercede.
it’s a horrible situation to be in and I wish it wasn’t so. We had to coach FIL to say he couldn’t have MIL home from hospital and the fact she was now doubly incontinent was the clincher. She went straight to a nursing home. But she wasn’t violent or aggressive, and she had clearly lost all capacity by that point.