left bounded chain complex $Z$ such that $Z \otimes_R^{\mathbf L} (R_P/PR_P)$ is uniformly right bounded as $P$ varies over prime ideals of $R$

Question

Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $Z$ be a left bounded chain complex of finitely generated $R$-modules. If there exists an integer $n$ such that for every prime ideal $P$ of $R$, the derived tensor-product $Z \otimes_R^{\mathbf L} (R_P/PR_P)$ is homologically right bounded upto $n$, then is $Z$ itself homologically bounded?

I feel like this must be some sort of Nakayama kind of argument, but I am not sure.

Please help.