The higher-rank charge-statistic conjecture for type $C_3$ crystals

Let $\mathcal{B}$ be a crystal of type $C_3$, let $\mathcal{B}_+(\lambda)$ denote the relevant positive part, and let $\epsilon_i$ and $s_i$ be the crystal statistics and simple reflections. A charge statistic is the statistic referred to in the source. Higher-rank charge-statistic conjecture. There exists a function $\widehat{\epsilon}_{32}:\mathcal{B}\rightarrow\mathbb{Z}_{\geq 0}$ such that

$$c(T)=\epsilon_1(T)+\epsilon_2(T)+\epsilon_2(s_1(T))+\epsilon_3(T)+\epsilon_3(s_2(T))+\epsilon_3(s_1s_2(T)) +\widehat{\epsilon}_{32}(T)+\widehat{\epsilon}_{32}(s_1(T))+\widehat{\epsilon}_{32}(s_2s_1(T))$$

is a charge statistic on $\mathcal{B}_+(\lambda)$. The conjecture further predicts that when $\operatorname{wt}(T)=0$, one has $c(T)=\epsilon_1(T)+2\epsilon_2(T)+3\epsilon_3(T)+3\widehat{\epsilon}_{321}(T)$. The claim proposes an extension of the charge statistics constructed in type $C_2$ to higher rank; no resolution is supplied in the source.