Power determinantal representation conjecture for real zero polynomials

Let $p\in\mathbb{R}[x_1,\ldots,x_n]$ be a real zero polynomial if, for every $x\in\mathbb{R}^n$ and $\mu\in\mathbb{C}$, $p(\mu x)=0$ implies that $\mu$ is real. Suppose that $p(0)=1$. Power determinantal representation conjecture. There are symmetric matrices $A_1,\ldots,A_n$ and a positive integer $N$ such that

$$p(x_1,\ldots,x_n)^N=\det(I+x_1A_1+\cdots+x_nA_n).$$

This is a relaxation of the Helton–Vinnikov conjecture, allowing a positive power of the polynomial. It is refuted in this paper using the nonrepresentability of the Vámos cube matroid, which has the half-plane property.