NTP2 conjecture for Kaplansky algebraically maximal valued fields

Let $(K,v)$ be a valued field of characteristic $p>0$ that is Kaplansky and algebraically maximal, and let $k$ be its residue field.

Valued-field NTP2 conjecture. Then $(K,v)$ is $\operatorname{NTP}_{2}$ (respectively, strong) if and only if $k$ is $\operatorname{NTP}_{2}$ (respectively, strong).

This asks for a positive-characteristic analogue of preservation results for $\operatorname{NTP}_{2}$ and related tameness properties under valuation-theoretic constructions; its resolution is not given here.