Eigenvalue realization conjecture for locally singular matrices in $\mathcal{S}^{n,k}$

Let $H(e^n_k)$ denote the hyperbolicity cone of the elementary symmetric polynomial $e^n_k$, and let $DG(n,k)D$ be a diagonal congruence of the matrix $G(n,k)$ by a diagonal matrix $D$. A vector $\lambda$ has at most one negative entry if at most one of its coordinates is negative. Eigenvalue realization conjecture. If $\lambda\in H(e^n_k)$, $e^n_k(\lambda)=0$, and $\lambda$ has at most one negative entry, then $\lambda$ is a vector of eigenvalues for $DG(n,k)D$ for some diagonal matrix $D$. The conjecture asks whether these necessary spectral conditions are also sufficient for realizing the eigenvalue vector of a nonsingular locally singular matrix in $\mathcal{S}^{n,k}$; the supplied text does not indicate that it has been resolved.