Eventual vanishing of $\operatorname{Ext}^i_R(M,N)$ Vs. eventual vanishing of $\operatorname{Ext}^i_R(M,R)$, where $N$ has projective dimension $2$

Question

Let $(R,\mathfrak m)$ Noetherian local reduced ring. Let $N$ be a finitely generated $R$-module of projective dimension $2$ ,i.e. there exists positive integers $a,b,c$ and an exact sequence $0\to R^{\oplus a}\xrightarrow{f} R^{\oplus b} \xrightarrow{g} R^{\oplus c} \to M \to 0 $ such that $f$ and $g$ have entries in $\mathfrak m$. If $M$ is a finitely generated $R$-module such that $\operatorname{Ext}^i_R(M,N)=0$ for all large enough $i\gg 0$, then is it true that $\operatorname{Ext}^i_R(M,R)=0$ for all large enough $i\gg 0$ ?