Coefficient-positivity conjecture for the multivariate affine type $B$ Eulerian polynomial

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

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

Here $\mathbb{N}[\mathbold{x},\mathbold{y}]$ denotes the polynomials in $\mathbold{x}$ and $\mathbold{y}$ with nonnegative integer coefficients. Coefficient-positivity conjecture.

$$\widetilde{B}_n(\mathbold{x},\mathbold{y})\in\mathbb{N}[\mathbold{x},\mathbold{y}]$$

for $n\geq 2$. The source notes that this multivariate refinement appears to be monomial positive, but gives no proof, so the conjecture remains open.