The conjectural induction rule for big partitions with two conormal nodes

Let $\tilde\lambda$ be the regularization of a big partition $u$ of $n$ having height $H$, and let $\bar\lambda_1$ and $\overline{h(\lambda)}$ denote the relevant residues. Let $\alpha\in\mathbb Z_p$ satisfy $\overline{\lambda_1}=-\overline{h(\lambda)}=\alpha$ not holding. Suppose that $\tilde\lambda$ has more than one conormal node of residue $\alpha$, and let $A$ and $B$ be those nodes, with $A$ above $B$. Write $\mathcal H_\varepsilon$ for the partition operation indexed by the displayed sequence $\varepsilon$.

Induction branching conjecture. There are exactly two such conormal nodes, and

$$[\operatorname{Ind}^\alpha D^{\tilde\lambda}]= \begin{cases} 2[D^{\tilde\lambda^A}]+[D^{\tilde\lambda^B}],&h_{2,1}(\lambda)\ne p-1,\\ 2[D^{\tilde\lambda^A}]+[D^{\tilde\lambda^B}]+[D^{\mathcal H_{(-2,2)}(\tilde\lambda^A)}],&h_{2,1}(\lambda)=p-1\text{ and }H=2,\\ 2[D^{\tilde\lambda^A}]+[D^{\tilde\lambda^B}]+[D^{\mathcal H_{(0,-1,1)}(\tilde\lambda^A)}]+[D^{\mathcal H_{(1,-1,0)}(\tilde\lambda^A)}],&h_{2,1}(\lambda)=p-1\text{ and }H=3,\\ 2[D^{\tilde\lambda^A}]+[D^{\tilde\lambda^B}]+[D^{\mathcal H_{(0,-1,0^{H-3},1)}(\tilde\lambda^A)}]+[D^{\mathcal H_{(0,-1,0^{H-4},1,0)}(\tilde\lambda^A)}],&h_{2,1}(\lambda)=p-1\text{ and }H>3. \end{cases}$$

The bracketed terms denote composition-factor multiplicities.

The conjecture is based on the upper estimates and calculations in known decomposition matrices. The supplied parser marks it as disproved, so it is recorded as refuted despite the conjectural wording in the paper.