Sokal's total monomial positivity conjecture for Hadamard products of dual Jacobi–Trudi matrices

Let $\mathbf{x}=(x_1,x_2,\dots)$ and $\mathbf{y}=(y_1,y_2,\dots)$ be distinct sequences of variables, and let

$$M(\mathbf{x})=\bigl(e_{j-i}(\mathbf{x})\bigr)_{i,j\geq 0},$$

where $e_k$ denotes the elementary symmetric function. For skew shapes $\lambda/\mu$, define the dual Jacobi–Trudi matrix

$$\mathcal{E}_{\lambda/\mu}(\mathbf{x})=\bigl(e_{\lambda_i-\mu_j-i+j}(\mathbf{x})\bigr)_{i,j=1}^{\ell(\lambda)}.$$

Sokal's conjecture. The Hadamard product $M(\mathbf{x})*M(\mathbf{y})$ is totally monomial positive; equivalently, every minor

$$\det\bigl(\mathcal{E}_{\lambda/\mu}(\mathbf{x})*\mathcal{E}_{\lambda/\mu}(\mathbf{y})\bigr)$$

is a multi-symmetric function in $\mathbf{x}$ and $\mathbf{y}$ with nonnegative integer coefficients. This strengthens Maló's total-positivity theorem by asking for coefficientwise positivity rather than positivity after specializing the variables to nonnegative real numbers. The source does not state whether the conjecture has been resolved.