The $q$-multiplicity duality conjecture for root systems of type $C$

Let $U_{\lambda,\mu}(q)$ be the $q$-analogue of the multiplicity of the representation $V(\lambda)$ in $\frak{H}^{C_m}(\mu)$, and let $X_{\lambda,\mu}(q)$ be the corresponding crystal-theoretic $q$-multiplicity defined using the combinatorial $R$-matrix and energy function for the crystals $B_{\mu_1}\otimes\cdots\otimes B_{\mu_m}$. For partitions $\lambda$ and $\mu$ of length $m$, write $|\lambda|$ and $|\mu|$ for their sizes. The $q$-multiplicity duality conjecture. For any partition $\lambda$ and $\mu$ of length $m$ with $|\mu|\geq|\lambda|$,

$$U_{\lambda,\mu}(q)=q^{|\mu|-|\lambda|}X_{\lambda,\mu}(q).$$

This conjecture predicts that the two $q$-analogues of the same representation multiplicity differ only by the explicit power of $q$ shown above. It is presented as suggested by computations; no resolution is supplied in the source.