Hal Schenck kindly provided a counter-example to the claim in question.
Consider the twisted cubic: (the code is in Macaulay)
                   2                        2
o3 I = ideal (- y  + x*z, - y*z + x*w, - z  + y*w)

i4 : J = ideal leadTerm I

              2        2
o4 = ideal (z , y*z, y )

i5 : hilbertPolynomial coker gens ideal leadTerm I^2

o5 = - 16*P  + 9*P
            0      1

i6 : hilbertPolynomial coker gens ideal leadTerm J^2

o6 = - 20*P  + 10*P
            0       1
So, the quotient is actually positive dimensional.