If $\mathbf{D}$ is the complex unit disc with coordinate function $s$ and $X \to \mathbf{D}$ is a proper flat holomorphic family (and it is smooth outside of the fiber $s=0$), will the total family $X$ deformation retract onto the fiber above $s=0$?

I don't believe this will be true, but I cannot find a counterexample. My idea was that if $X$ is a family of elliptic curves degenerating to a singular cubic, then this would fail (but, I think it actually works in this case!).

Edit: in the comments, the consensus is that this should be true - but we do not have a proof yet.