Triviality over a finite invariant base for non-definable invariant types

Let $T$ be a countable weakly quasi-o-minimal theory with few countable models. Let $\mathcal P$ be the collection of all global $1$-types that are invariant over some finite parameter set but are not definable. Finite-base triviality conjecture. For every $\mathfrak p\in\mathcal P$, there is a finite set $A$ such that $\mathfrak p$ is $A$-invariant and $\mathfrak p_{\restriction A}$ is trivial. This conjecture is one of two proposed steps toward confirming Martin's conjecture for $T$; its general status is left open in the source.