Strong pattern-avoidance conjecture for dual canonical basis immanants

Let $0<k<n\leq m$ be integers, let $v\in S_n$ avoid the pattern $12\cdots(k+1)$, and let $R,C\in\binom{[m]}{n}$, where repetitions are allowed. Strong pattern-avoidance conjecture. If $\operatorname{Imm}_v X(R,C)$ is not identically zero, then $\operatorname{Imm}_v X(R,C)$ is $k$-positive.

This is presented as an intermediate conjecture: it would imply Pylyavskyy's conjecture and would follow from Pylyavskyy's conjecture together with the sign-control conjecture. The supplied text does not report a resolution.