The refined $m$-shuffle conjecture for $\nabla^m E_{n,k}$

Fix positive integers $m,n$ and set $\delta_n=(n-1,n-2,\ldots,1,0)$. Let $E_{n,k}$ be defined by

$$e_n\left[Z\frac{1-u}{1-q}\right]=\sum_{k=1}^n\frac{(u;q)_k}{(q;q)_k}E_{n,k}(z),$$

and let $D_n^{(m),\lambda}(z;q)$ denote the generalized tableau sum associated with a partition $\lambda\subseteq m\delta_n$. The refined $m$-shuffle conjecture. For $1\leq k\leq n$,

$$\nabla^m E_{n,k}=\sum_{\substack{\lambda\subseteq m\delta_n\\|\{i:\lambda_i=m(n-i)\}|=k}}t^{|m\delta_n/\lambda|}D_n^{(m),\lambda}(z;q).$$

This is proposed as the $m>1$ analogue of the refinement for $\nabla E_{n,k}$, separating the contributions according to the number of parts on the boundary of $m\delta_n$. The supplied text gives no resolution.