The conjectural height-three induction rule

Let $\lambda$ be a completely splittable partition of height $3$ with $h_{1,1}(\lambda)=2p-1$. Let $\tilde\lambda$ denote its regularized partition, and let $A=(3,\tilde\lambda_3+1)$ and $B=(4,1)$ be the specified nodes. Let $\mathcal H_{(0,1,-1)}$ denote the corresponding partition operation.

Height-three induction conjecture.

$$[\operatorname{Ind}^{-\bar 3}D^{\tilde\lambda}]=2[D^{\tilde\lambda^A}]+[D^{\tilde\lambda^B}]+[D^{\mathcal H_{(0,1,-1)}(\tilde\lambda^A)}].$$

This is a special branching formula for completely splittable partitions of height three. The source gives no proof or resolution status.