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Without further assumptions, no. As the other have already pointed out. The Gröbner basis of $I$ could also be $\langle x,y \rangle$ which gives the same $LT(I)$
I've always wondered if there is a faster way to get the $n$ ideals $I:x_i$ than computing $n$ elimination orders from scratch. There should be some Gröbner walk type technique that lets one modify the elimination order GB for one variable to get to the next. Is that what your reference is about?
