"5000 each to other siblings to buy the house outright?"

But the house belonged in the estate, as the Whole. You have the $4k each; not $5k. He didn't buy the house from each of them, one at a time.

That's the difference between the house going into an estate, and the house being owned outright by 5 people and one will buy out the other 4.

"TP invests 20000 into the estate"

I doubt they had him/her improve the property until the estate no longer owned it; and the money (an asset) would have removed the house (an asset) from the estate. That's why people are confused over the selling of a $62k FMV for $20k.

"TP share of sale is 31570"

You cannot tell from the data here that there was any loss, because no one here has the improvement invested value in the consideration of the details.

It seems you need to confirm who is the owner of this property at each stage of the process. Either it is part of the estate or someone owned it outright.