Hurwitz stability conjecture for the peak polynomial sequence

Let $M_{n,r}(x)$ and $\widetilde{M}_{n,r}(x)$ be the polynomial sequences defined by the preceding peak-polynomial recurrences, and define $T_{n,r}(x)$ by

$$(1+x)T_{n,r}(x)=xM_{n,r}(x^2)+\widetilde{M}_{n,r}(x^2).$$

Hurwitz stability conjecture. The polynomial $T_{n,r}(x)$ is Hurwitz stable for all $r\ge 2$ and $n\in\mathbb{N}$. The sequence is introduced after Hurwitz stability has been established for the associated peak-polynomial sequences and their Turán expressions. The conjecture is based on empirical evidence and computer arithmetic, and its resolution is not specified in the source.