Casselman--Shahidi conjecture on the singularity of intertwining operators: groups of exceptional type

As a sequel to our recent work on Casselman--Shahidi's holomorphicity conjecture on half-normalized intertwining operators for quasi-split classical groups, we modify our method, based on a lemma of Heiermann--Opdam, to prove certain cases of the conjecture for groups of exceptional type uniformly. One main ingredient, established here, is to find an algorithm to produce reduced decompositions of co-rank one relative longest Weyl elements, in terms of certain "small" counterparts.