Let $R$ be a commutative Noetherian ring, and $D^b(\operatorname{mod } R)$ the bounded derived category of the abelian category of finitely generated $R$-modules. Let me abbreviate this as $D^b(R)$. Consider the singularity category $D_\text{sg}(R)$ as the Verdier localization $D^b(R)/{\operatorname{perf}(R)}$, where $\operatorname{perf}(R) :=\operatorname{thick}_{D^b(R)}(R)$ is the subcategory of $D^b(R)$ of perfect complexes.

My question is the following: Given $M,N\in D^b(R)$, is it true that $M\in \operatorname{thick}_{D_\text{sg}(R)}(N)$ if and only if $M\in \operatorname{thick}_{D^b(R)}(N\oplus R)$ ?