NTP2 conjecture for pseudo real closed and pseudo p-adically closed fields

Recall that a field is pseudo real closed (PRC) if every absolutely irreducible variety defined over it that has a rational point in every real closure has a rational point in the field. A field is pseudo p-adically closed (PpC) if every absolutely irreducible variety defined over it that has a rational point in every p-adic closure has a rational point in the field. A field is bounded if, for every integer $n$, it has only finitely many Galois extensions of degree $n$.

PRC and PpC NTP2 conjecture. A PRC field is $\operatorname{NTP}_{2}$ if and only if it is bounded. Similarly, a PpC field is $\operatorname{NTP}_{2}$ if and only if it is bounded.

The unbounded PRC case is known to have $\operatorname{TP}_{2}$, while the corresponding characterization for PRC and PpC fields remains conjectural.