Schilling–Shimozono conjecture for the branching-component automorphism

Let $D$ be a $\pm$ diagram of shape $\Lambda/\lambda$ whose ambient $r\times s$ rectangle has complement of $\Lambda$ tiled by vertical dominos. Let $m_{i}^{\circ}$, $m_{i}^{+}$, $m_{i}^{-}$ and $m_{i}^{\pm}$ count columns of height $i$ having respectively no symbol, a $+$, a $-$, or a $\pm$ pair. If $r$ is even, define $m_{0}^{\circ}=s-\Lambda_{1}$. For $i\equiv r+1\pmod 2$, all $m_i^*$ with $*\in\{\circ,+,-,\pm\}$ vanish. Schilling–Shimozono conjecture. The automorphism $\check{\sigma}(D)$ is the $\pm$ diagram with, for every $0\leq i\leq r$, $m_{i+2}^{\pm}$ empty columns of height $i$, $m_i^-$ columns of height $i$ carrying a $+$, $m_i^+$ columns of height $i$ carrying a $-$, and $m_i^\circ$ columns of height $i+2$ carrying a $\pm$ pair. This gives the conjectural shape-preserving automorphism of the branching component graph used to construct the affine crystal automorphism.