Lusztig varieties for regular elements

Let $G$ be a connected reductive group over an algebraically closed field. Let $B$ be a Borel subgroup of $G$ and $W$ be the associated Weyl group. We show that for any $w \in W$ that is not contained in any standard parabolic subgroup of $W$, the intersection of the Bruhat cell $B w B$ with any regular conjugacy class of $G$ is always irreducible. We then prove that the associated Lusztig varieties are irreducible. This extends the previous work of Kim \cite{kim2020homology} on the regular semisimple and regular unipotent elements. The irreducibilitiy result of Lusztig varieties will be used in an upcoming work in the study of affine Lusztig varieties.