Critical-resonance conjecture for failure of McNamara's vanishing

Fix a long word $\underline{i}$ and a dominant coweight $\lambda$. Let ${\bf m}$ be an $\underline{i}$-Lusztig datum, let $I_\lambda({\bf m})$ be its Whittaker integral, and let $\mathcal{B}(\lambda+\rho)$ be the highest-weight crystal. A datum is critically resonant when it exhibits the critical resonance described by the paper.

Critical-resonance conjecture. If

$$I_\lambda({\bf m})\neq0\quad\text{and}\quad(\underline{i},{\bf m})\notin\mathcal{B}(\lambda+\rho),$$

then ${\bf m}$ is critically resonant.

This conjecture predicts that critical resonance completely accounts for exceptions to McNamara's proposed vanishing criterion. The authors state that a uniform evaluation for resonant data is not currently known and present this as a direction for future work.