Laver's well-order conjecture for Hurwitz-action domains

Let $(S, \mathbin{*})$ be a left-cancellative LD-system, let $B_n$ be the braid group on $n$ strands, and let $D_S(\vec a)=\{\beta\in B_n\mid \vec a\mathbin{\scriptscriptstyle\bullet}\beta\text{ is defined}\}$ be the domain of the partial Hurwitz action on a sequence $\vec a\in S^n$. A free LD-system is an LD-system freely generated by its specified generators. Laver's conjecture. If $(S,\mathbin{*})$ is a free LD-system, the restriction of the D-ordering to every family of the form $D_S(\vec a)$ is a well-order. This extends Laver's well-order result for positive braids; the conjecture remains open when $\vec a$ has length $3$ or more.