Stable multivariate refinement conjecture for affine type $B$ Eulerian polynomials

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

$$\widetilde{B}_n(\mathbold{x},\mathbold{y})=2\widetilde{C}_n(\mathbold{x},\mathbold{y})-2nx_ny_nB_{n-1}(\mathbold{x},\mathbold{y};1).$$

Write $\mathfrak{S}_{\mathbb{R}}[\mathbold{x},\mathbold{y}]$ for the class of real stable polynomials in the variables $\mathbold{x}$ and $\mathbold{y}$. Stable-refinement conjecture.

$$\widetilde{B}_n(\mathbold{x},\mathbold{y})\in\mathfrak{S}_{\mathbb{R}}[\mathbold{x},\mathbold{y}]$$

for $n\geq 2$. The polynomials were verified by computer to be stable for $n\leq 4$, and the conjecture proposes a stable multivariate refinement of the affine type $B$ Eulerian polynomial.