The conjectural restriction rule for completely splittable partitions

Let $\lambda$ be a completely splittable partition of height $H$ with $\chi(\lambda)=p$, and let $\alpha\in\mathbb Z_p$. Let $A$ be the top $\lambda$-addable node, namely the node in the first row. Suppose that $\lambda^A$ has more than one normal node of residue $\alpha$, and let $B$ be the bottom $\lambda$-removable node. Let $x=[h_{2,1}(\lambda)\geqslant p]$ be the indicator of the stated inequality.

Restriction branching conjecture. Except for $A$, the only $\lambda^A$-normal node of residue $\alpha$ is $B$, and

$$[\operatorname{Res}_\alpha D^{\lambda^A}]= \begin{cases} 2[D^{\tilde\lambda}]+[D^\lambda]+[D^{\mathcal H_{(-1,0^{H-3},1,0)}(\lambda)}]+x[D^{\mathcal H_{(0,-1,0^{H-3},1)}(\lambda)}],&H>2,\\ 2[D^{\tilde\lambda}]+[D^\lambda]+x[D^{\mathcal H_{(1,-1)}(\lambda)}],&H=2. \end{cases}$$

This is a proposed restriction-side branching formula, based on upper estimates and known decomposition matrices. The source supplies no proof or resolution status.