The diagonal-monomial reachability conjecture for psd-stable polynomials

Let $f$ be a psd-stable polynomial in symmetric variables, and let $Z^\beta$ be a monomial appearing in a psd-stable polynomial. A diagonal monomial is a monomial $Z^\alpha$ appearing in $f$ whose variables are diagonal variables. The distance between monomials is measured by the norm used for exponent vectors, and linear, double, and transposition steps are the support-preserving monomial moves defined in the paper.

Diagonal-monomial reachability conjecture. For any monomial $Z^\beta$ appearing in a psd-stable polynomial, there is a diagonal monomial $Z^\alpha$ appearing in $f$ which can be reached by a sequence of linear, double and transposition steps which decrease the distance from $\beta$ to $\alpha$ and which never leave the support of $f$.

This conjecture proposes that the determinant's connectedness under support-preserving combinatorial moves extends to all psd-stable polynomials. Its resolution is not specified in the supplied text.