Canonical positive monomial expression conjecture for non-commutative Schur functions

Let $\lambda$ be a partition of size $k$, and let $s_\lambda({\mathbf u})$ and $h_\lambda({\mathbf u})$ be the non-commutative Schur and homogeneous symmetric functions in the ribbon Schur operators $u_i$. A monomial is written as $\underline{u}=u_{i_k}u_{i_{k-1}}\cdots u_{i_1}$. Canonical expression conjecture. The functions $s_\lambda({\mathbf u})$ can be written as a non-negative sum of monomials, with a canonical expression obtained by selecting monomials occurring in $h_\lambda({\mathbf u})$; specifically, the selected monomials satisfy the strict inequalities within each consecutive block of indices determined by the parts of $\lambda$:

$$i_{\lambda_1}>i_{\lambda_1-1}>\cdots>i_1,$$ $$i_{\lambda_1+\lambda_2}>i_{\lambda_1+\lambda_2-1}>\cdots>i_{\lambda_1+1},$$

and so on. This is proposed as the paper's main problem and would provide a canonical combinatorial explanation for the non-negativity of the associated skew $q$-Littlewood–Richardson coefficients.