Let $R$ be a ring. An $R$-module $M$ is called FL (FP) if it has a finite resolution consisiting of finitely generated free (projective) modules.
Given an exact sequence of $R$-modules, $0\to M_1\to M_2\to M_3\to 0$, if two of the modules $M_1$, $M_2$ or $M_3$ are FP, then the third is FP as well. This follows, for example, from Proposition 1.4 and Proposition 4.1b of
Bieri, Robert Homological dimension of discrete groups. Second edition. Queen Mary College Mathematics Notes. Queen Mary College, Department of Pure Mathematics, London, 1981.
My question is
If two of the modules $M_1$, $M_2$ or $M_3$ are FL, is it true that the third one is also FL?