Let $(R,\mathfrak m)$ be a local Cohen-Macaulay ring of dimension $n$ with a canonical module $\omega$. Let $M$ be a finitely generated $R$-module with $\text{depth } M=\dim M=t$. Using Bruns&Herzog's book, Cohen-Macaulay rings, Corollary 3.5.11 (a consequence of Grothendieck local-duality), I can see that $\text{Ext}^i_R(M,\omega)\ne 0$ if and only if $i=n-t$ and $\dim \text{Ext}^{n-t}_R(M,\omega)\le n-(n-t)=t.$

My question is: Is the module $\text{Ext}^{n-t}_R(M,\omega)$ Cohen-Macaulay? i.e., is it true that $\text{depth } \text{Ext}^{n-t}_R(M,\omega)=\dim \text{Ext}^{n-t}_R(M,\omega)$?

Commutative algebraX, §9, no. 1, Corollaire of Proposition 3. $\endgroup$