Coefficient-positivity conjecture for the multivariate type $D$ Eulerian polynomial

For $n\geq 2$, let $D_n(\mathbold{x},\mathbold{y})$ be the multivariate refinement defined by

$$D_n(\mathbold{x},\mathbold{y})=B_n(\mathbold{x},\mathbold{y};1)-n2^{n-1}x_ny_nA_{n-2}(\mathbold{x},\mathbold{y}).$$

Here $A_{n-2}$ and $B_n$ are the stable multivariate refinements of the type $A$ and type $B$ Eulerian polynomials, respectively. Coefficient-positivity conjecture.

$$D_n(\mathbold{x},\mathbold{y})\in\mathbb{N}[\mathbold{x},\mathbold{y}]$$

for all $n\geq 2$. The conjecture was verified by computer for $n\leq 11$; if true, it could yield a refinement of the type $D$ descent statistic and a new recursion for type $D$ Eulerian polynomials.