Quadratic generation from monomial-freeness of $J_1$

Let $J_1$ and $J_2$ be the ideals under consideration, with $J_1$ monomial-free. Quadratic-generation conjecture. If $J_1$ is monomial-free, then $J_2$ is quadratically generated. This claim concerns the relationship between monomial-freeness and quadratic generation in the ideals arising from the paper's standard monomial and toric degeneration constructions; the supplied text gives no evidence resolving it.