Let (R,m) be a Cohen-Macaulay local ring, with canonical module ω, and let I be an ideal of R.

Question:  Is the minimal number of generators of $Hom_R(ω, Iω)$  equal to the minimal number of generators of I?

The answer is affirmative  if R is Gorenstein.

Does anybody know an answer in the general case, please?

Thank you so much.