Computational conjecture on rex moves with a common rightmost generator

Let $W$ be the Coxeter group, let $s\in S$, and let $\underline{w}$ and $\underline{w'}$ be reduced expressions, each ending in $s$. A rex move is a morphism between reduced expressions generated by braid relations. Rex-move computational conjecture. For any rex move $\beta:\underline{w}\rightarrow\underline{w'}$ with both expressions having $s$ on the far right, the diagrammatic relation displayed in the source holds. The source says this would imply, using the stated diagrammatic relations, that a subsequent diagram is zero; no general proof or disproof is given.