Exhaustion conjecture for indecomposable parabolic homomorphisms of irreducible crystallographic Hecke monoids

Let $(W(M),\star)$ and $(W(M'),\star)$ be Hecke monoids of irreducible crystallographic types. A homomorphism is injective parabolic if it is both injective and parabolic, and it is indecomposable as a homomorphism of Hecke monoids if it cannot be decomposed nontrivially in that category. The homomorphisms described above are the injective parabolic homomorphisms obtained from the preceding constructions. Exhaustion conjecture. The homomorphisms described above exhaust, up to diagram automorphisms and decorations, all injective parabolic homomorphisms between Hecke monoids of irreducible crystallographic types except for $G_2$ that are indecomposable as homomorphisms of Hecke monoids. This would classify the indecomposable injective parabolic homomorphisms in the stated types, modulo the indicated equivalences; the source supplies no resolution status.