Lower bound for characteristic variety

Let K be an algebraically closed field of char. 0, let A_n(K) be the Weyl algebra. Let I in A_n(K) be a left ideal generated by p elements. Set M := A_n / I.

Does the following then hold?

dim Ch(M) \geq 2n - p

Here Ch(M) is the characteristic variety of M. (I know that the answer is yes if n = 1, and also if I is generated by homogeneous elements.)

Thank you.