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 $V$ of $X\times H^0(X, E^*\otimes F)$ parametrizing the degeneracy loci $D_k(\phi)=\{x \in X \; |\; \mathrm{rk}(\phi_x)\leq k \}$, i.e. $V=\{(x,\phi)\;|\; x\in D_k(\phi)\}$. Is $V$ Cohen-Macaulay?