The conjectural $A_7$-crystal decomposition of $B^{1,s}$

Let $B^{1,s}$ be the Kirillov–Reshetikhin crystal of type $E_7^{(1)}$, and let $\eta_i$ denote the fundamental weights of $A_7$. For $a,b,c,d\in\mathbb{Z}_{\geq 0}$ satisfying $a+2b+3c+d\leq s$, define

$$m_{d,s'}=\sum_{i=M}^{d+1}\left\lceil\frac{i}{2}\right\rceil,$$

where $M=\max(d+1-(s'-d),0)$, and set $m_{a,b,c,d}=m_{d,s-a-2b-3c}$. The conjectural $A_7$-crystal decomposition. As $A_7$ crystals,

$$B^{1,s}\cong\bigoplus B\bigl(a(\eta_1+\eta_7)+b(\eta_2+\eta_6)+c(\eta_3+\eta_5)+d\eta_4\bigr)^{\oplus m_{a,b,c,d}}.$$

This gives the proposed decomposition of $B^{1,s}$ into $A_7$ crystals. The paper notes that the $A_7$ diagram symmetry is compatible with the $E_7^{(1)}$ diagram symmetry, and that proving the decomposition via the $E_7$-crystal decomposition could potentially prove the cited conjecture of Joseph and Shimozono; no proof or resolution is supplied here.