Stable Schur-positivity conjecture for nabla applied to $[200~\operatorname{\Xi}e_\mu$

Let $[200~\mu$ be a partition, let $[200~\boldsymbol{x}$ be a set of variables, and let $[200~\operatorname{\oldXi}$ be the operator appearing in the expression below. Let $[200~\boldsymbol{\nu}$ range over lexicographically ordered sequences of partitions whose first parts are all larger than $1$, and let $E_{\boldsymbol{\nu}}([200~\mathcal{X})$ denote the corresponding basis elements. Stable coefficient Schur-positivity conjecture. There exist stable coefficients $c_{[200~\mu,\boldsymbol{\nu}}(q,t)$, Schur-positive polynomials in $q,t$, such that

$$\left.\nabla_{\mathcal{Y}}(\operatorname{\oldXi}e_\mu(\boldsymbol{x}))\right|_{q\to q+1,\,t\to t+1}=\sum_{l\geq0}\sum_{\ell(\boldsymbol{\nu})=l}c_{\mu,\boldsymbol{\nu}}(q,t)E_{\boldsymbol{\nu}}(\mathcal{X}).$$

The conjecture extends the observed positivity of explicit expansions after the substitutions $q\mapsto q+1$ and $t\mapsto t+1$; the supplied context gives calculations but does not state a proof or resolve the claimed stability.