Lascoux–Leclerc–Thibon's Schur-positivity conjecture for ribbon-tableau functions

Let $\mathcal G^{(n)}_\lambda(q,x)$ be the symmetric functions arising from ribbon tableaux and Fock-space representations of the quantum affine algebra $U_q(\widehat{\mathfrak{sl}}_n)$. For a partition $\lambda$ and integers $1\leq i\leq n$, define

$$\lambda^{[i,n]}=(\lambda_i,\lambda_{i+n},\lambda_{i+2n},\ldots).$$

For symmetric functions $f$ and $g$, $f\geq_s g$ means that $f-g$ is Schur nonnegative. Lascoux–Leclerc–Thibon's conjecture. For integers $1\leq m\leq n$ and a partition $\lambda$,

$$\prod_{i=1}^n s_{\lambda^{[i,n]}}\geq_s\prod_{i=1}^m s_{\lambda^{[i,m]}}.$$

The claim is the $q=1$ reformulation of the Lascoux–Leclerc–Thibon conjecture for the family $\mathcal G^{(n)}_\lambda(q,x)$. The source does not specify a resolution status.