A question over Gorenstein local rings

Question

Let $(R,{\frak m})$ be a Gorenstein local ring and $M$ and $N$ be two indecomposable finitely generated $R$-modules such that $\dim M= \dim N=\dim R=n$. Suppose that $\operatorname{Hom}_R(M,R)‎\cong‎ \operatorname{Hom}_R(N,R)\neq 0 $ and $H^n_{\frak m}(M) \cong H^n_{\frak m}(N)$ (where $H^n_{\frak m}(-)$ denotes the $n$-th right derived functor of $\Gamma _{\frak m}(-)$). Can we conclude that $M \cong N$?

Answer 1

These assumptions are still not enough. Let $\dim R=n\geq 2$ and let $I$ be an ideal generated by a maximal regular sequence. Then both $I,R$ are indecomposable, their duals are just $R$ and $H^n_{\mathfrak{m}}(I)=H^n_{\mathfrak{m}}(R)$.