The composition-factor multiplicity conjecture for Kac modules of $\mathfrak{gl}(m/n)$

Let $\mu$ be integral dominant and $r$-fold atypical with respect to the roots $\gamma_1<\gamma_2<\cdots<\gamma_r$ ($\gamma_i\in\Delta_{1,+}$), and let $k_i=\#\nabla(\gamma_i)$. For $\theta=(\theta_1,\ldots,\theta_r)\in\{0,1\}^r$, define

$$\lambda_\theta={\dot d}(\mu_\theta)={\dot d}\left(\mu+\sum_{i=1}^r\theta_i k_i\gamma_i\right).$$

The composition-factor multiplicity conjecture. For each of the $2^r$ integral dominant weights $\lambda_\theta$, $a_{\lambda_\theta,\mu}=1$, and $a_{\lambda,\mu}=0$ for every other weight $\lambda$.

This gives a simple description of the composition factors of the Kac module associated with an integral dominant atypical weight. The claim is presented as the paper's main result, but the supplied material does not establish its resolution.