Lecouvey's charge formula conjecture for type $C_n$ Kostka–Foulkes polynomials

Let $\beta$ be a dominant weight for the root system $C_n$, let $B(\beta)$ be the corresponding crystal, and let

$$B(\beta)_\mu=\{T\in B(\beta):\operatorname{wt}(T)=\mu\}$$

be its subset of tableaux of weight $\mu$. Let $\operatorname{ch}_n(T)$ denote the charge statistic defined on symplectic tableaux, and let $K_{\beta,\mu}(q)$ be the Kostka–Foulkes polynomial for type $C_n$. Lecouvey's charge conjecture. For $\lambda,\mu\in P_n^+$,

$$K_{\lambda,\mu}(q)=\sum_{\mathrm{w}(T)\in B(\lambda)_\mu}q^{\operatorname{ch}_n(T)}.$$

The charge is proposed as a type $C_n$ analogue of Lascoux–Schützenberger charge, giving a tableau formula for Kostka–Foulkes polynomials. The paper notes that the formula is supported by computations and proves the special case corresponding to $k=1$ in the preceding comparison conjecture, but does not establish the general statement.