Let $X$ denote a generic $n \times m$ (with $n \leq m$) matrix and $R = k[X]$, where $k$ is any field. Let $J := I_n (X)$. It is well-known that $J^t = J^{(t)}$ for all $t$ (where $-^{(t)}$ denotes the symbolic power). My question is the following: Is it also known that if $I := \textrm{in}_< (J)$ for any term order, then $I^t = I^{(t)}$ for every $t$?

It is also known that $I^t = \textrm{in}_< (J^t)$ for every $t$, and also when $n=1$ or $2$ the answer to my question is yes (the case $n=1$ is clear, and for $n=2$ the initial ideal is a facet ideal of a bipartite graph, which is Mengerian, and this is known to be equivalent to my question).

My suspicion is that my question is well-known and probably follows from something more general, but I am not as well-versed as I should be with the literature on symbolic powers. Any help is appreciated!



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