Injectivity criterion for dihedral standard homomorphisms into type A

Fix $n>1$, let $\operatorname{Br}^+_{n+1}=\operatorname{Br}^+(A_n)$, and let $\sigma$ be the diagram automorphism with $\sigma(i)=n+1-i$ for $i\in[1,n]$. For an interval $K\subsetneq[1,n]$ with $|K|>1$, let $[\Phi](1)$ and $[\Phi](2)$ denote the supports of a homomorphism $\Phi:\operatorname{Br}^+(I_2(N))\to\operatorname{Br}^+_{n+1}$ as in the converse of the paper's classification theorem; a subset is self-orthogonal in the sense of that theorem.

Dihedral injectivity criterion. Both induced maps $\overline\Phi$ and $\overline\Phi_\star$ are injective if and only if either both $[\Phi](1)$ and $[\Phi](2)$ are self-orthogonal, or $K$ satisfies $\sigma(K)=K$.

This criterion is a consequence of the paper's classification of optimal disjoint fully supported standard homomorphisms from dihedral type into type $A$. Its status is unclear from the supplied metadata because the parser marked the conjecture environment as unresolved.