The normalized inertia conjecture for partial Hessians of symmetric polynomials

Let $q$ be the partial Hessian of a symmetric polynomial, let $Z(0,A)$ be the associated middle matrix evaluated at $A\in\mathbb S_n(\mathbb R^g)$, let $\mu_\pm$ denote the number of positive or negative eigenvalues, and let $\sigma_\pm(q)$ denote the corresponding positive or negative signature parameter. Normalized inertia conjecture.

$$\sup_n\left\{\frac{\mu_\pm(Z(0,A))}{n}: A \in \mathbb S_n(\mathbb R^g) \right\}=\sigma_\pm(q).$$

The surrounding results establish this equality under additional hypotheses, including that $q$ is the partial Hessian of a symmetric polynomial of degree at least three and that $\sigma_\pm(q)\leq 1$; the unrestricted formulation stated here is not resolved in the supplied text.