Let $X$ be a Cohen-Macaulay projective variety (say over an algebraically closed field of characteristic 0), and $E,F$ two vector bundles such that $E^*\otimes F$ is globally generated. Consider the subscheme $V$ in $X\times H^0(X, E^*\otimes F)$ parametrizing the degeneracy loci $D_k(\phi)=\{x \in X \; |\; \mathrm{rk}(\phi_x)\leq k \}$, for $\phi \in H^0(X, E^*\otimes F)$. Is $V$ Cohen-Macaulay?