Conjecture on Hurwitz actions and reduced decompositions of braid Coxeter elements

In either of the two settings, let $B$ be the associated braid group, let $W$ be the corresponding Coxeter or complex reflection group, let $p:B\twoheadrightarrow W$ be the natural map, and let $R$ and $T$ be respectively the sets of braid reflections and reflections. Let $n$ be the rank, let $B_n$ act by Hurwitz moves, and let $\operatorname{Red}_R(g)$ and $\operatorname{Red}_T(c)$ denote the reduced decompositions of $g$ and $c$ into elements of $R$ and $T$. For a braid Coxeter element $g$, write $c:=p(g)$, and let $R_g$ and $T_c$ be the reflections appearing in these reduced decompositions. The reduced-decomposition conjecture. There exists a braid Coxeter element $g\in B$ such that: (1) the Hurwitz action is transitive on $\operatorname{Red}_R(g)$; (2) the Hurwitz action is transitive on $\operatorname{Red}_T(c)$; (3) the map $p^n$ induces an isomorphism of $B_n$-sets from $\operatorname{Red}_R(g)$ to $\operatorname{Red}_T(c)$; and (4) the map $p$ induces a bijection from $R_g$ to $T_c$. The conjecture is proved for the universal Coxeter group and braid group, and substantial cases are known for finite Coxeter groups and affine type $\widetilde{A_n}$; the general statement, including the complex-reflection-group setting, remains open.