This question concerns the following Lemma 4.2 in this paper by Van den Bergh:
Lemma: Let $R$ be local, normal Gorenstein ring of dimension $d$. Suppose $M$ is a reflexive $R$ module such that $A=\text{End}_R(M)$ is maximal Cohen-Macaulay (i.e. $A$ has maximum depth, in this situation $A$ is often called an $R$-order). Suppose $\text{gl.dim} (A)<\infty$, then $\text{gl.dim}(A)=d$.
My question is, does the Gorenstein hypothesis necessary for dimension $d>2$:
Is the above Lemma true with $R$ being normal and Cohen-Macaulay, assuming $d>2$?
and
If not, is there a upper bound on $\text{gl.dim} (A)$? ($d$ is an obvious lower bound)
(The proof in quoted paper used Gorensteiness in the second sentence: "$\text{Ext}^i_R(A,R)=0$ for $i>0$..." Under mild hypotheses one can try to replace $R$ by the canonical module, but then run into troubles in line 3...).
