Generalized determinantal representation conjecture for real-zero polynomials

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 $x^0$. Generalized determinantal representation conjecture. There exists an $RZ_{x^0}$ polynomial $h \in {\mathbb R}[x_1,\ldots,x_d]$ of degree $\ell$ with $h(x^0)=1$, such that the closure of the connected component of $x^0$ in $\{x \in {\mathbb R}^d \colon h(x)>0\}$ contains the closure of the connected component of $x^0$ in $\{x \in {\mathbb R}^d \colon p(x)>0\}$, and there are matrices $A_0,A_1,\ldots,A_d \in {\mathbb S}^{n\times n}$, with $n\geq m+\ell$ and

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

such that

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

This is proposed as the best possible generalization of the planar LMI representation theorem to higher dimensions: a determinantal representation may require an additional real-zero factor $h$, while the resulting matrix size is bounded below by the total degree $m+\ell$.