Just to be clear, in the scenario I posted, there is no full or partial share of the house being inherited. The estate has an asset which is to be sold, the proceeds of the estate being split 4 ways.
The question was "when unceretainty exists, why not value it high for probate, thus avoiding having to pay tax on any increase in value locked in on sale", i.e. no estate tax if sale price >= probate price.
There are many examples of people inheriting houses valued pre-crash in 2007/8 and then selling the house a year later at a greatly reduced price, but having to pay tax on the probate value. They did, of course, have huge losses to carry forward against future gains