Let $R$ be a DVR and $S = \mathrm{Spec}(R)$ with closed point $s$ and generic point $\eta$. Let $X\to S$ be a scheme over $S$, and denote by $X_s$ and $X_{\eta}$ the special and generic fibers. We assume that there exists a finite number of closed points $x_1,\ldots ,x_n \in X_s$ such that the open subscheme $U := X \setminus \{x_1,\ldots,x_n\}$ is smooth over $S$.
Let $\ell$ be a prime number different from the residual characteristic of $R$. Let $\Lambda := \mathbb Z_{\ell}/ \ell^k\mathbb Z_{\ell}$ where $k\geq 1$. Consider the derived nearby cycle functor $\mathrm R\Psi_{\eta}$ relative to $X$.
Is it true that $\mathrm R^0\Psi_{\eta}\Lambda \simeq \Lambda$ and $\mathrm R^q\Psi_{\eta}\Lambda$ for $q\geq 1$ is skyscraper concentrated in the singular points $x_1,\ldots ,x_n$?
Does this generalize to $\mathrm R\Psi_{\eta}K$ for $K \in D^+(X_{\eta},\Lambda)$ any bounded-below complex of $\Lambda$-adic sheaves on $X_{\eta}$?