The diagonal-rescaled elementary symmetric spectral containment conjecture

Let $D$ be a positive definite diagonal matrix, and let $p(x)=e_k(Dx)$. Diagonal-rescaled elementary symmetric conjecture. The polynomial $p(x)$ has the spectral containment property. This is a concrete special case of the broader spectral containment problem for homogeneous multiaffine stable polynomials; the supplied context does not state whether this special case is known or open.