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

Fix $n$ and let $\delta_n=(n-1,n-2,\ldots,1,0)$. For each partition $\lambda\subseteq\delta_n$, let $D_n^\lambda(z;q)$ denote the tableau sum over semistandard tableaux of shape $(\lambda+(1^n))/\lambda$, weighted by $q^{\operatorname{dinv}(T)}z^T$. Let $E_{n,k}$ be defined by

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

The refined shuffle conjecture. For $1\leq k\leq n$,

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

This refinement isolates the tableaux indexed by partitions having exactly $k$ parts on the staircase boundary and implies the stated positivity consequence for $\nabla E_{n,k}$. The supplied text introduces it as a refinement of the main conjecture and gives no resolution.