Gaetz–Pierson non-negativity conjecture for increasing-pattern coefficients

Let $v=12\dots k$ be the increasing permutation pattern, let $N_{12\dots k}$ count occurrences of this pattern in permutations, and let $\operatorname{ch}_n(RN_{12\dots k})$ denote its Schur expansion. For a fixed partition $\lambda$, write $\lambda[n]$ for the associated partition indexing the Schur function $s_{\lambda[n]}$. Gaetz–Pierson's conjecture. The coefficient of $s_{\lambda[n]}$ in

$$\operatorname{ch}_n(RN_{12\dots k})$$

is non-negative. This conjecture extends the known non-negativity results for the coefficients indexed by $(n-2,1,1)$ and $(n-2,2)$ in the increasing-pattern case.