Algebraic Geometry – Does a Resolution of a Rational Singularity Have Rationally Connected Fibers?

Question

A rational singularity is a singularity of a
complex variety $X$ such that for any
resolution $\pi:\; \tilde X\rightarrow X$ the
higher direct images $R^i\pi_*(O_{\tilde X})$
vanish for all $i>0$. Suppose that $X$
has isolated rational singularity,
and $\tilde X\rightarrow X$ its resolution.
I expect that the fiber of $\pi$ over
the singular point is rationally
connected; I would be very grateful
for any reference to this. I need to apply
this to the local situation, so it would be
especially nice if the argument does not
use projectivity.

Answer 1

No. For instance the cone over an Enriques surface (with respect to any projective embedding) has rational singularity, but Enriques surface is not rationally connected.