Monov's conjecture on moments of critical points of realizable lists

Let $\Lambda=\{\lambda_1,\dots,\lambda_n\}\subset\mathbb{C}$ with $n\ge 2$ be a list, let

$$p(z)=\prod_{i=1}^n(z-\lambda_i),$$

and let $\Lambda'=\{\mu_1,\dots,\mu_{n-1}\}$ be the list of zeros of $p'$. A list is realizable if it is the spectrum of an entrywise nonnegative matrix, and write

$$s_k(\Lambda')=\sum_{i=1}^{n-1}\mu_i^k.$$

Monov's conjecture. If $\Lambda$ is realizable, then

$$s_k(\Lambda')\ge 0\quad\text{for all }k\in\mathbb{N}.$$

The conjecture proposes an additional necessary condition for the nonnegative inverse eigenvalue problem, concerning the critical points of the characteristic polynomial of a realizable list. Its resolution is not indicated in the supplied text.