The generalized Lax conjecture in algebraic form

Let $p$ be a polynomial hyperbolic with respect to $e\in \mathbb{R}^n$, and let $\Lambda_+(p,e)$ denote its hyperbolicity cone. A polynomial has a definite determinantal representation if it can be represented by the determinant of a linear matrix pencil that is positive definite at $e$. Generalized Lax conjecture. There exists a polynomial $q$, hyperbolic with respect to $e\in \mathbb{R}^n$, such that $qp$ has a definite determinantal representation and

$$\Lambda_+(q,e)\supseteq \Lambda_+(p,e).$$

This is the algebraic formulation of the geometric generalized Lax conjecture, which asks whether every closed hyperbolicity cone is spectrahedral. The original trivariate Lax conjecture is a theorem of Helton and Vinnikov, but the corresponding statement in higher dimensions remains open.