Isometric dimension conjecture for braid graphs of reduced expressions

Let $(W,S)$ be a simply-laced triangle-free Coxeter system, and let ${\boldsymbol{\alpha}}$ be a reduced expression with link factorization

$${\boldsymbol{\alpha}}_1\mid {\boldsymbol{\alpha}}_2\mid\cdots\mid {\boldsymbol{\alpha}}_k.$$

Write $B({\boldsymbol{\alpha}})$ for its braid graph, let $\dim_I$ denote isometric dimension, and let $\operatorname{rank}({\boldsymbol{\alpha}}_i)$ denote the rank of each link factor.

Reduced-expression isometric-dimension conjecture.

$$\dim_I(B({\boldsymbol{\alpha}}))=\sum_{i=1}^k\operatorname{rank}({\boldsymbol{\alpha}}_i).$$

This conjecture is presented as a consequence of the link dimension conjecture together with the product decomposition of braid graphs; it remains conditional and unproved in the supplied text.