Classification conjecture for fully supported standard A2 homomorphisms

Let $I$ be a Coxeter graph, let $M\in\mathbf{Cox}(I)$ be irreducible of finite type, and let $\Phi\in\operatorname{Hom}_{\operatorname{\mathscr{Art}}}(A_2,M)$ be fully supported and standard, with $[\Phi](1)\ne[\Phi](2)$.

A2 classification conjecture. Up to diagram automorphisms, either $M=A_{3m-1}$ for some $m\ge1$ and $\Phi=\Phi_{m,J}$ with $J\subset[1,m-1]$ as in Theorem~, or $M=E_6$ and $\Phi=\widehat{\Phi}_{3,E_6}$ as in Proposition~.

The paper lists this among conjectural families of non-disjoint standard homomorphisms and reports verification for $n\le15$. It is intended as an exhaustive classification up to diagram automorphisms.