Uglov crystal characterization conjecture for weak Harish-Chandra series

Let $\lambda\in\mathcal{P}$, let $t\in\mathbb{N}$ satisfy $\lambda_{(2)}=\Delta_t$, and set $\mu=\bar{\lambda}^{(2)}$. Assume that $\ell$ is sufficiently large, that $e$ is odd, and put $\mathbf{c}=(t+(1-e)/2,0)$. Let $|\mu,\mathbf{c}\rangle$ denote the corresponding vertex of the crystal graph $\mathcal{G}_{\mathbf{c},e}$. Uglov conjecture. The module $X_\lambda$ is weakly cuspidal if and only if $|\mu,\mathbf{c}\rangle$ is a source vertex of $\mathcal{G}_{\mathbf{c},e}$. If $X_\lambda$ is weakly cuspidal and $\rho\in\mathcal{P}$, then $X_\rho$ lies in the weak Harish-Chandra series defined by $X_\lambda$ if and only if $\rho_{(2)}=\lambda_{(2)}=\Delta_t$ and $|\bar{\rho}^{(2)},\mathbf{c}\rangle$ lies in the connected component of $\mathcal{G}_{\mathbf{c},e}$ containing $|\mu,\mathbf{c}\rangle$; equivalently, it is obtained from that vertex by adding a sequence of good nodes. This would turn weak Harish-Chandra cuspidality and series membership into combinatorial questions on the crystal graph, assuming the preceding graph comparison. The source gives no resolution.