Schur-positivity conjecture for triangular-partition step sequences

Let $[200~\boldsymbol{v}$ be a vector that is the step sequence of a triangular partition, and let $[200~\mathcal{E}_{\boldsymbol{v}}(\boldsymbol{x})$ be the symmetric function defined from the associated rational weights and modified Macdonald polynomials. Let $[200~\mathcal{X}$ denote the combined variable sets on which $[200~\nabla_{\mathcal{Y}}$ acts. Triangular-step-sequence positivity conjecture. Whenever $[200~\boldsymbol{v}$ is the step sequence of a triangular partition, $[200~\nabla_{\mathcal{Y}}(\mathcal{E}_{\boldsymbol{v}}(\boldsymbol{x}))$ is $[200~\mathcal{X}$-Schur positive. Moreover, its $e$-expansion in all the $[200~\mathcal{Y}$-variables after $q\mapsto q+1$ and $t\mapsto t+1$ has Schur-positive coefficients in $q,t$. The claim generalizes the explicit positivity phenomena studied in the paper and is motivated by the cited terminology for triangular partitions; no resolution is given in the supplied text.