Noncommutative Schur positivity modulo the $k$-R relations

Let $\boldsymbol{u}$ be the generators of the noncommutative algebra ${\mathcal{U}}$, let $\mathfrak{J}_\lambda(\boldsymbol{u})$ be the noncommutative Schur function associated to an integer partition $\lambda$, and let $I_{\mathrm{R},k}$ be the ideal generated by the stated $k$-dependent relations. For $\gamma\in {\mathcal{U}}^*_{\ge 0}\cap I_{\mathrm{R},k}$, let $F_\gamma(\mathbf{x})$ denote the associated symmetric function.

Noncommutative Schur positivity conjecture. For every integer partition $\lambda$ and every positive integer $k$, $\mathfrak{J}_\lambda(\boldsymbol{u})$ is $\mathbb{Z}$-monomial positive modulo $I_{\mathrm{R},k}$. Consequently, every $F_\gamma(\mathbf{x})$ with $\gamma\in {\mathcal{U}}^*_{\ge 0}\cap I_{\mathrm{R},k}$ is Schur positive.

This statement is inspired by work of Assaf and would imply Schur positivity for the associated family of symmetric functions, including the LLT and transformed Macdonald examples described in the paper. Its resolution is not supplied in the source.