Hurwitz Equivalence of Reflection Factorizations in Types $B$ and $D$

A conjecture of Lewis asserts that two reflection factorizations of the same element in a finite Coxeter group $W$ belong to the same Hurwitz orbit if and only if they generate the same subgroup $H$ of $W$ and have the same multiset of $H$-conjugacy classes. We prove this conjecture for every finite Coxeter group all of whose irreducible components are of type $A$, $B$ or $D$. Furthermore we prove a reduction, which reduces the conjecture to a finite problem for the exceptional types.