Hayashi's quiver quotient conjecture for finite-dimensional weak bialgebras

Let $\kk$ be an algebraically closed field. A finite-dimensional weak bialgebra with commutative counital subalgebras is a finite-dimensional weak bialgebra $H$ whose counital subalgebras are commutative. For a finite quiver $Q$, let $\hay$ denote the weak bialgebra attached to $Q$ by Hayashi.

Hayashi's quiver quotient conjecture. Every finite-dimensional weak bialgebra $H$ with commutative counital subalgebras is isomorphic to a weak bialgebra quotient of $\hay$ for some finite quiver $Q$.

This conjecture proposes a quiver presentation framework for quantum symmetries of algebras with commutative separable degree-zero components. It is stated as a modification of a result of Hayashi and remains unproved.