What happens if my ex continually refuses to be bought out of the home?

Location: New York

Hello. I live in New York State. I’m going through the process of a divorce. Thankfully, I won my dog in court, with the judge citing the reason as not wanting to remove the dog from the home she has lived in for the last five years- my home. My ex and I purchased the house under joint tenancy in 2020, before the market exploded. We were not married at the time, and I paid a very hefty down payment on my own, from my personal account. From then on, until about 2023, he sent me sporadic Venmo payments. Sometimes he would pay the entire mortgage’s worth, other times he sent me nothing for months. Every mortgage payment ultimately came out of my separate personal account. (We never co-mingled bank accounts.) I also paid for all repairs, several of which were before we were married in 2022, some were after we married.

In late 2023, I caught him cheating and he left the home. He now lives with the affair partner in an apartment. He cannot afford to buy me out or take over the mortgage. I can afford to buy him out, because the mortgage didn’t change much in our 3 years living together, and the down payment came out of my account alone. But of course, since the housing market changed so much since 2020, there is still quite a lot of equity he’s now fighting for (the home has appreciated in value by around 120k).

If it’s ruled marital property, only some of my repairs could be subtracted from his equity. If it’s ruled separate, all of them could be removed and he’d get very little. But my question comes down to this: what if he refuses ALL buy out offers? Even extremely fair ones that reflect market value? I have a feeling that even if I offered him the full half, 60k, right now, he’d refuse. He is extremely bitter and angry that I won the dog, and he wants to force the sale to force me from my home. Could he do this? Just refuse to accept all offers to force me out?