My question is problem 2.9 of Gröbner Bases in Commutative Algebra by Ene & Herzog. If $\mathfrak m$ is the graded (or homogeneous) maximal ideal of polynomial ring $S$, the Loewy length of $S/I$ is the infimum of the numbers $k$ such that $\mathfrak m^k\subseteq I$.
The question is to prove $S/I$ and $S/\mathrm{in}_<(I)$ have the same Loewy length.
I can prove that if $\mathfrak m^k\subseteq I$ then $\mathfrak m^k\subseteq\mathrm{in}_<(I)$, but I can't show the other side. Thank you for any hint about this.