Hurwitz-stability conjecture for higher Toeplitz determinant polynomials

Let $PF_r$ be the class of sequences whose infinite Toeplitz matrix has all minors of order at most $r$ non-negative. For $r\geq2$, define

$$P_n^r(x):=\sum_{k_1+\cdots+k_r=n}\binom{n}{k_1,\ldots,k_r}f_{k_1}\cdots f_{k_r}\det\bigl((x-i+j)_{k_i}\bigr)_{1\leq i,j\leq r}.$$

Here $(x)_k$ is the rising factorial, and Hurwitz stability means that all zeros have negative real part. The higher-order Hurwitz-stability conjecture. If $\{f_k\}_{k=0}^{n}\in PF_r$, then $P_n^r(x)$ is Hurwitz stable.

This is the stability counterpart to the preceding degree and coefficient-positivity conjecture. Its status is not resolved in the supplied source context.