Farey-interval shape conjecture for Mullineux wall-crossing of the sign representation

Let $n$ be a prime number, let $\operatorname{sgn}_n$ be the partition of $n$ corresponding to the sign representation of $S_n$, and let $r$ be a term in the $n$-th Farey sequence. For $s=p/q$, let $M_s$ be the $q$-Mullineux involution on partitions of $n$, let $0<w_1<\dots<w_k<r$ be the terms of the $n$-th Farey sequence between $0$ and $r$, and define

$$N_r=M_r\circ M_{w_k}\circ\dots\circ M_{w_1}.$$

For positive integers $f_1<\dots<f_m$ and $g_1,\dots,g_m$, write $f_1^{g_1}\dots f_m^{g_m}$ for the partition having $g_i$ parts equal to $f_i$. Farey-interval shape conjecture. There is an increasing sequence of fractions $a_i/b_i$ in the Farey series such that, whenever $a_i/b_i<r<a_{i+1}/b_{i+1}$, the partition $N_r(\operatorname{sgn}_n)$ has the form $z^t x^y$, where

$$z+y=b_i$$

and

$$y+t+z-x=b_{i+1}.$$

Together with $xy+zt=n$, these equations constrain the shape of the wall-crossed sign representation. The source presents this as a computer-motivated conjecture and gives no resolution.