Real-negative-zero conjecture for Pólya-frequency Toeplitz polynomials

Let $\{f_k\}_{k=0}^{n}$ be a Pólya frequency sequence of infinite order, denoted $PF_{\infty}$, meaning that all minors of the associated infinite Toeplitz matrix are non-negative. For $n\geq3$, define $Q^{1,1}_n(x)$ as above. The real-negative-zero conjecture. If $\{f_k\}_{k=0}^{n}$ is $PF_{\infty}$ and $n\geq3$, then all zeros of $Q^{1,1}_n(x-1)$ are real and negative.

This is presented as a new conjecture in the source and is intended as a stronger, real-rooted analogue of the preceding stability claim. Its status is not resolved in the supplied source context.