The hook-restricted Schur expansion conjecture for the nabla operator

Let $n$ be a positive integer, let $abla$ be the \nabla operator on symmetric functions, and let $e_n$ and $s_{\mu}$ denote the elementary and Schur symmetric functions, respectively. For a partition $\mu\vdash n$, write $\operatorname{SYT}(\mu)$ for the set of standard Young tableaux of shape $\mu$, and for $\tau\in\operatorname{SYT}(\mu)$ let $\operatorname{maj}(\tau)$ and $\operatorname{des}(\tau)$ denote its major index and descent number. Let $s_{a,b}(q,t)$ denote the corresponding two-variable Schur function, and let $\left.\langle\nabla(e_n),s_{\mu}\rangle\right|_{\operatorname{hooks}}$ denote the hook-restricted part of the scalar product. The hook-restricted Schur expansion conjecture. For all $\mu\vdash n$,

$$\left.\langle\nabla(e_n),s_{\mu}\rangle\right|_{\operatorname{hooks}}=\sum_{\tau\in\operatorname{SYT}(\mu)}s_{\operatorname{maj}(\tau)}(q,t)+\sum_{i=2}^{\operatorname{des}(\tau)}s_{\operatorname{maj}(\tau)-i,1}(q,t).$$

The formula has been established in the preceding argument for the cases treated there, and the authors state that they conjecture it for all partitions $\mu$ when $m=1$. They also note that the analogous assertion is false for $m>1$.