Lattice conjecture for prefixes of a braid Coxeter element

Let $B$ be the associated braid group, let $R$ be its set of braid reflections, and let $B_+$ be the submonoid of $B$ generated by $R$. Define a relation $\preccurlyeq$ on $B_+$ by

$$b\preccurlyeq b'\quad\Longleftrightarrow\quad b^{-1}b'\in B_+.$$

For $b\in B_+$, set

$$P_b:=\{b'\in B_+\mid b'\preccurlyeq b\}.$$

A braid Coxeter element is an element of $B$ as defined in the preceding setup. Lattice conjecture. There exists a braid Coxeter element $g\in B$ satisfying the reduced-decomposition conjecture and such that $(P_g,\preccurlyeq)$ is a lattice. This conjecture is intended to provide the key lattice property needed for the associated monoid to be quasi-Garside; its validity remains open in the general settings considered.