Are injective maps of finite modules over regular ring liftable to termwise injective maps of finite projective resolution?

Question

Let $R$ be a regular noetherian ring and let $M$ be a finitely generated $R$-module. In this situation, we know that $M$ admits a finite resolution in terms of finite projective $R$-modules.

Now, let $f:M\to M$ be an injective $R$-linear map. By projectivity, there is a lift of $f$ to the projective resolution of $M$.

My question is as follows:

Can I always find a finite resolution of $M$ by finite projective resolution of $M$, say $(M_i)_i$, and lifts $f_i:M_i\to M_i$ of $f$ in such a way that, for all $i$, $f_i$ is also injective?