Uniqueness conjecture for non-abelian simple A-H Frobenius monoids

Let ${\mathcal C}$ be a compact closed category, and let $N\cong X\otimes X^*$ be a self-dual object. The standard A-H Frobenius algebra at $N$ is the A-H Frobenius algebra supplied by the paper's preceding theorem. When $N$ is strictly reflexive, its endomorphism monoid is a semi-monoidal monoid and contains a standard A-H F. monoid; when ${\mathcal C}$ has trivial scalars, this is called a simple A-H F. monoid. A standard A-H Frobenius monoid is non-degenerate when it satisfies the paper's non-degeneracy condition.

Uniqueness conjecture. Non-degenerate standard A-H Frobenius monoids differ only by their scalars. More precisely, but less generally, all non-abelian simple A-H F. monoids are isomorphic.

This conjecture proposes uniqueness up to scalar data, with the explicitly stated stronger-looking special formulation for non-abelian simple A-H F. monoids. The supplied text does not establish whether either formulation is resolved.