Type $C_n$ versus type $A_{n-1}$ Kostka–Foulkes comparison conjecture

Let $\Lambda_k^A$ and $\Lambda_{n-k}^A$ be the $k$-th and $(n-k)$-th fundamental weights of $U_q(\mathfrak{sl}_n)$, and set

$$\lambda_k=\Lambda_k^A+\Lambda_{n-k}^A.$$

Let $K_{\Lambda_{2k},0}(q)$ be the type $C_n$ Kostka–Foulkes polynomial and $K^A_{\lambda_k,0}(q)$ the type $A_{n-1}$ polynomial for weight $0$. Type $C_n$–type $A_{n-1}$ comparison conjecture. One has

$$K_{\Lambda_{2k},0}(q)=K^A_{\lambda_k,0}(q^2).$$

This comparison motivates the proposed charge formula by identifying a relevant type $A_{n-1}$ crystal component inside the type $C_n$ crystal. The paper states that the conjecture is true for $k=1$, while the general equality is left as a conjecture.