EL-shellability of exceptional quasi-Coxeter intervals in types E7 and E8

Prove that the interval [1,w] in the absolute order is EL-shellable for every quasi-Coxeter element w of a finite irreducible Coxeter group, in particular for the unresolved classes E_7(a_3), E_7(a_4), and E_8(a_2),E_8(a_8).

Background

The paper proves EL-shellability for all quasi-Coxeter intervals except possibly those in types E_7 and E_8. The authors conjecture that the natural phenomenon extends universally, but their computations do not resolve nine exceptional classes: E_7(a_3), E_7(a_4), and E_8(a_2) through E_8(a_8). These intervals are substantially larger than the computed H_4 cases, preventing the authors from determining whether they are EL-shellable.

For the unresolved classes, no total order of the reflections is compatible with the natural edge labeling, so any eventual EL-labeling must use a different labeling. Establishing shellability would complete the main shellability result and confirm the stated conjecture.