Hurwitz stability conjecture for the alternating-run polynomial

Let $r(x)=\sqrt{\frac{1+x}{1-x}}$ and define $\mathscr{R}_n(x)=\sum_{k=0}^{2n-1}\mathscr{R}(n,k)x^k$ by

$$(xD_x)^n(r(x))=\frac{\mathscr{R}_n(x)}{(1-x)^n(1+x)^{n-1}\sqrt{1-x^2}}.$$

The recurrence is

$$\mathscr{R}_{n+1}(x)=(2nx+1)x\mathscr{R}_n(x)+x(1-x^2)D_x\mathscr{R}_n(x),$$

with $\mathscr{R}_0(x)=1$ and $\mathscr{R}_1(x)=x$. Hurwitz stability conjecture. The polynomial $\mathscr{R}_n(x)$ is Hurwitz stable for $n\in\mathbb{N}$. The coefficients count dual Stirling permutations by alternating runs, and the polynomial is not real-rooted; the conjecture asserts the stronger half-plane location property instead. Its resolution is not specified in the source.