I would like to find a example of a graded ideal $I\subset k[x_1,\cdots,x_n]$ for which $\mbox{in}_<(I)$ is not even Cohen-Macaulay (for some monomial order).
I have tried to find such a ideal using Singular and Macaulay2, but until to now I got nothing. Such example is asked in the book of Herzog & Higi, Monomial ideals.
I would be happy with any hint!