Let $f\colon X\to Y$ be a flat morphism of irreducible projective algebraic varieties over $\mathbb{C}$ (or any other algebraically closed field of characteristic 0). Assume that $Y$ is smooth, and the generic fiber of $f$ is smooth (these two assumptions seem to be important).

Let $U\subset Y$ be a Zariski open non-empty subset such that $f\colon f^{-1}(U)\to U$ is smooth. Fix a closed point $y\in Y$.

Let us denote by $S$ the "disk", more precisely $S$ is the spectrum of henselization of $\mathbb{C}[t]$ localized at the ideal generated by $t$. Let $\eta$ be the generic point of $S$.

For any morphism $\nu\colon S\to Y$ such that $\eta$ is mapped to $U$ and the closed point of $S$ to $y$, consider the fibered product $S\times_Y X\to S$. Notice that the generic fiber of this morphism is smooth over $\eta$. Consider the nearby cycle functor of the constant sheaf $\underline{\mathbb{\mathbb{Q}_l}}$. It is a perverse sheaf on $f^{-1}(y)$ which we will denote by $\mathcal{F}_\nu$ to emphasize dependence on the morphism $\nu$.

QUESTION.Is it true that for all choices of $\nu$ as above the perverse sheaves $\mathcal{F}_\nu$ are isomorphic to each other?