Derived tensor product of finitely generated module of finite projective dimension and bounded above chain complex of finitely generated free modules

Question

Let $(R,\mathfrak m)$ be a reduced Noetherian local ring. Let $0\neq G$ be a finitely generated $R$-module of finite projective dimension. Let $M$ be a bounded above chain complex of finitely generated free $R$-modules.

If $M \otimes_R^{\mathbf L} G$ is homologically bounded, then is $M$ also homologically bounded?