Higher-power nabla conjecture for q,t-rectangle numbers

For $a,b\geq 0$, let $S_{a,b}(q,t)$ be the q,t-rectangle numbers defined from words with $a$ letters $\mathrm{N}$ and $b$ letters $\mathrm{E}$, and let $p_n$ and $s_{(1^n)}$ denote the power-sum and Schur symmetric functions. Let $\nabla$ be the nabla operator and $\langle\,\cdot\,,\,\cdot\,\rangle$ the Hall inner product. Higher-power nabla conjecture. For all $m\geq 0$ and $n>0$,

$$(qt)^{mn(n-1)/2}S_{n,mn}(1/q,1/t)=(-1)^{n-1}(m+1)\langle\nabla^m(p_n),s_{(1^n)}\rangle.$$

This generalizes the proved square case to rectangles of dimensions $n\times mn$ and relates their specialized generating functions to higher powers of nabla. The source presents the general statement as conjectural.