Let $R$ be a commutative Noetherian ring and $\mod R$ be the (abelian) category of finitely generated $R$-modules. Let $\mathcal D^b(\mod R)$ be the bounded derived category of $\mod R$. Each finitely generated $R$-module $M$ can be thought of as a member of $\mathcal D^b(\mod R)$ by regarding it as a chain complex concentrated in degree zero.
For $X\in \mathcal D^b(\mod R)$, let thick $ X$ be the intersection of all thick subcategories (https://ncatlab.org/nlab/show/thick+subcategory) of $\mathcal D^b(\mod R)$ containing $X$.
My question is : Let $R$ be a local ring such that for every finitely generated $R$-module $M$, the subcategory thick $M$ of $\mathcal D^b(\mod R)$ contains a non-exact perfect complex (depending on $M$ of course). Then, is it true that every finitely generated $R$-module $M$ embeds in a finitely generated $R$-module of finite projective dimension?