Let $M,N$ be finitely generated modules over a commutative Noetherian local ring $(R, \mathfrak m)$. Assume that $\text{Ext}^i_R(M,N)=0$ for all large integers $i\gg 0$, and also that $N$ has projective dimension $1$ ,i.e., there exists an exact sequence $0\to R^{\oplus a} \xrightarrow{f} R^{\oplus b}\to N \to 0$ where $a,b$ are positive integers and $\text{Im}(f)\subseteq \mathfrak m R^{\oplus b}$.
Then, is it true that $\text{Ext}^i_R(M,R)=0$ for all large integers $i\gg 0$ ?