non-free module of finite projective dimension and locally free on punctured spectrum on a local non-Cohen-macaulay ring

Question

Consider the local ring $R=\mathbb C [[x,y,z]]/(xy, yz)$. My question is: Does there exist a non-free finitely generated $R$-module $M$ of finite projective dimension such that $M_P$ is $R_P$-free for every non-maximal prime ideal $P$ of $R$?

Thoughts: I think $R$ has dimension $2$ and depth $1$. Moreover, I believe $R_P$ is regular local every non-maximal prime ideal $P$ of $R$. I still have no idea how to construct a module as in the question.

Answer 1

Take $M$ to be the quotient, $0\to R\stackrel{(x,y,z)}{\to} R^3\to M\to 0$.