Steep-and-shallow criterion for generalized regularization and Mullineux commutation

Let $a$ and $b$ be positive integers with $2a<b$, and let $u$ be a partition that is both $\operatorname{Reg}_{a,b}$-valid and $\operatorname{Colreg}_{a,b}$-valid. For each hook $H_{i,j}$ of $u$ with $b\mid H_{i,j}$, let $l_{i,j}$ and $a_{i,j}$ denote its leg and arm lengths. Steep-and-shallow criterion conjecture. If every such hook satisfies either

$$\left(\frac{b}{a}-1\right)l_{i,j}<a_{i,j}+1$$

or

$$\left(\frac{b}{a}-1\right)a_{i,j}<l_{i,j}+1,$$

then

$$\nu^{\operatorname{Reg}_{a,b}\operatorname{M}_b}=\nu^{\operatorname{T}\operatorname{Reg}_{a,b}},$$

equivalently,

$$\nu^{\operatorname{Reg}_{a,b}\operatorname{M}_b\operatorname{T}}=\nu^{\operatorname{Colreg}_{a,b}}.$$

If $a$ and $b$ are coprime and $\nu$ satisfies this equality, then the steep-and-shallow hook condition in the source holds for all hooks divisible by $b$. The paper presents this as a conjectural characterization relating generalized regularization, column regularization, transposition, and the Mullineux involution.