An $m$-positivity conjecture related to bivariate Jacobi-Trudi matrices

Question

This question by Rellek reminds me of the following problem. Alan Sokal has conjectured that if we replace the elementary symmetric function $e_k$ in the dual Jacobi-Trudi matrix for $s_\lambda$ (a Schur function) by $e_k(x)e_k(y)$, then the determinant is $m$-positive, i.e., if we expand it in terms of the basis $m_\lambda(x)m_\mu(y)$ (where $m$ denotes a monomial symmetric function), then the coefficients are nonnegative. (See the 14:22 point of this video of a talk by Sokal.) Equivalently, let $f$ be the homomorphism from symmetric functions in one set $z$ of variables to functions that are symmetric in two sets $x$ and $y$ of variables separately defined by $f(e_n(z))=e_n(x)e_n(y)$. Then Sokal's conjecture is equivalent to $f(s_\lambda)$ being $m$-positive. Recently I made the stronger conjecture that $f(m_\lambda)$ is $m$-positive.