The classification conjecture for fully supported standard homomorphisms from $A_2$

Let $M\in\mathbf{Cox}(I)$ be irreducible of finite type, and let $\Phi\in\operatorname{Hom}_{\mathscr A}(A_2,M)$ be fully supported and standard with $[\Phi](1)\ne[\Phi](2)$. Classification conjecture for $A_2$ homomorphisms. Up to diagram automorphisms, either $M=A_{3m-1}$ for some $m\ge1$ and $\Phi=\Phi_{m,J}$ for $J\subset[1,m-1]$, or $M=E_6$ and $\Phi=\widehat\Phi_{3,E_6}$. The paper presents this as one of several conjectural families and reports verification for $n\le15$; no general proof or disproof is supplied here.