The hyperbolic Horn conjecture for lpm-polynomials

Let $P$ be any degree $k$ lpm-polynomial, and let $\lambda,\mu\in\mathbb{R}^k$ with $\lambda$ majorizing $\mu$. Hyperbolic Horn conjecture. There exists a symmetric matrix $X$ such that the roots of $P(X-tI)$ are given by $\lambda$, while the roots of $P(\operatorname{diag}(X)-tI)$ are given by $\mu$. This is proposed as a hyperbolic generalization of Horn's theorem, and the authors state that they have no results toward it.