Symmetric nonnegative matrix rank-one perturbation D-stability conjecture

Let $H\foral R^{n\times n}$ be symmetric and entrywise nonnegative, with geometrically simple spectral-radius eigenvalue $\rho(H)$. Let $v,w\in\mathbb R^n$ satisfy $v>0$ and $w>0$. Symmetric rank-one perturbation conjecture. The matrix

$$\rho(H)I-H+v w^T$$

is D-stable. A preceding counterexample shows that the analogous assertion without symmetry is false, while this symmetric case is raised as an open direction.