Fortuna's entrywise nonnegativity conjecture for Lyapunov solutions with companion state matrices

Let

$$a(s)=s^n+a_{n-1}s^{n-1}+a_{n-2}s^{n-2}+\dots+a_1s+a_0$$

be a Hurwitz polynomial, and let $A\in\mathbb{R}^{n\times n}$ be its companion matrix,

$$A=\begin{bmatrix}0&1&0&\dots&0\\0&0&1&\ddots&0\\vdots&\ddots&\ddots&\ddots&\ddots\\0&0&0&\dots&1\\-a_0&-a_1&-a_2&\dots&-a_{n-1}\end{bmatrix}.$$

Let $Q=Q^{\top}\in\mathbb{R}^{n\times n}$ satisfy $Q\succeq0$, and let $X$ be the unique solution of

$$XA+A^{\top}X=-Q.$$

Fortuna's conjecture. The solution $X$ is entrywise nonnegative.

Standard Lyapunov theory gives that $X$ is symmetric and positive semidefinite. The paper proves this claim when the companion matrix has only real eigenvalues, while the general assertion remains unresolved in the supplied text.