No broken diamond in hexagon conjecture for modified crystal operator paths

Let $S_{\lambda,\mu}$ be the set of Kostant partitions under consideration, and let $k_i$ and $k_{i+1}$ be the positive integers associated with $\boldsymbol\alpha\in S_{\lambda,\mu}$ as in Definition~. Suppose that the modified crystal operators $\phi_i$ and $\phi_{i+1}$ are both non-zero on $\boldsymbol\alpha$. Define the paths

$$E_i:=e_i^{k_{i+1}}e_{i+1}^{k_i+k_{i+1}}e_i^{k_i}(\boldsymbol\alpha),$$

and

$$E_{i+1}:=e_{i+1}^{k_i}e_i^{k_i+k_{i+1}}e_{i+1}^{k_{i+1}}(\boldsymbol\alpha).$$

No broken diamond in hexagon conjecture. The paths $E_i$ and $E_{i+1}$ in the graph induced by the modified crystal operators do not contain any edges that give way to Stembridge Exception $\rm{BD}$. This conjecture is the condition used to establish the stated braid relations for the modified operators; the source reports that it is supported by computer testing, but gives no proof or resolution.