Gurvits's derivative-determinantal representation conjecture

Let $p \in {\mathbb R}[x_1,\ldots,x_d]$ be an $RZ_{x^0}$ polynomial of degree $m$ with $p(x^0)=1$. Here, $RZ_{x^0}$ means real-zero with respect to $x^0$, and $q^{(k)}_{x^0}$ denotes the $k$th Renegar derivative of $q$ with respect to $x^0$. Gurvits's conjecture. There exist $k \in {\mathbb Z}_+$, an $RZ_{x^0}$ polynomial $q \in {\mathbb R}[x_1,\ldots,x_d]$ of degree $m+k$ such that $p=q^{(k)}_{x^0}$, and matrices $A_0,A_1,\ldots,A_d \in {\mathbb S}^{(m+k)\times(m+k)}$ with

$$A_0+x^0_1A_1+\cdots+x^0_dA_d=I,$$

such that

$$\det(A_0+x_1A_1+\cdots+x_dA_d)=q(x).$$

The conjecture offers an alternative way to generate real-zero polynomials by combining the two systematic constructions discussed in the source: positive real symmetric determinantal representations and Renegar derivatives. Its resolution would also clarify whether every such $p$ can be obtained as a derivative of a polynomial with a positive real symmetric determinantal representation.