Evans–Sturmfels–Uhler conjecture on commuting birth and death ideals

Let $E$ be a grid, and let $I^{E}$ denote its commuting birth and death ideal, formed as the sum of the ideals associated with all induced squares of $E$. Evans–Sturmfels–Uhler conjecture. For any grid $E$, the ideal $I^{E}$ is radical, its prime decomposition consists of pure toric ideals, and it is independent of the coefficient field. This conjecture is based on computations for small grids and concerns the structure and field-independence of these binomial ideals; the supplied source does not state whether it has been resolved.