Abstract generalized Lax conjecture for generalized Clifford algebras

Let $h$ be a hyperbolic polynomial in direction $e$, let $\mathcal A_e(h)$ be its generalized Clifford algebra, and let $\Sigma^2\mathcal A_e(h)$ denote the cone of sums of hermitian squares in that algebra. Let $\Lambda_e(h)$ be the hyperbolicity cone of $h$. Abstract generalized Lax conjecture. The hyperbolicity cone of $h$ should arise as a canonical linear section of the closed sums-of-squares cone in $\mathcal A_e(h)$. More precisely, for the canonical linear map $\iota\colon\mathbb R^n\to\mathcal A_e(h)$ defined by $a\mapsto a\bullet\sigma$, one should have

$$\Lambda_e(h)=\iota^{-1}\left(\overline{\Sigma^2\mathcal A_e(h)}\right).$$

Here the closure is taken in the finest locally convex topology and equals the double-dual cone. This is proposed as an abstract formulation of the generalized Lax conjecture; its resolution is not stated in the supplied text.