Let $R$ be a Commutative Noetherian ring, and let $\text{mod } R$ denote the abelian category of finitely generated $R$-module. Consider the bounded derived category $D^b(\text{mod } R) $ which is a triangulated category. Every object $X \in \text{mod } R$ can be naturally identified in $D^b(\text{mod } R) $. For an object $X \in D^b(\text{mod } R)$, let $\text{Thick}_{D^b(\text{mod } R)} X$ denote the intersection of all thick subcategories (https://ncatlab.org/nlab/show/thick+subcategory) of $D^b(\text{mod } R)$ containing $X$. For example, note that $\text{Thick}_{D^b(\text{mod } R)} R$ is the collection of all perfect complexes, hence $M \in \text{mod } R$ belongs to $\text{Thick}_{D^b(\text{mod } R)} R$ if and only if $M$ has finite projective dimension.
Now let $0\to Y \to P \to X \to 0$ be a short exact sequence in $\text{mod } R$ where $P$ is a finitely generated projective $R$-module. Then, is it true that $Y \in \text{Thick}_{D^b(\text{mod } R)} X$ ?