Tail Conjecture for highest weights of $gl(m|n)$ modules

Let $\lambda\in \Lambda^+_{m|n}$. Let $\Sigma_\lambda\in\mathbb{S}$ be the base for which $\Phi(\lambda)=D_\lambda^{\Sigma_\lambda}$, define $\lambda^{\dagger}:=\lambda_{\Sigma_\lambda}$ and $\overline{\lambda}^{\dagger}:=\lambda^{\dagger}+\rho_{\Sigma_\lambda}$, and set

$$tail(\lambda):=s\left(\overline{\lambda}^{\dagger}\right).$$

For the module with highest weight $\lambda$, write

$$longtail(\lambda):=\max_{\nu\in Hwt(\lambda)}s(\nu).$$

Tail Conjecture. For any $\lambda\in\Lambda^+_{m|n}$,

$$tail(\lambda)=longtail(\lambda).$$

The conjecture asserts that the tail determined by the distinguished weight diagram equals the maximum tail over the highest weights associated with the module. Its resolution is not specified in the supplied source material.