Finite-dimensional Frobenius algebra conjecture for multiplication-comultiplication axioms

Let $A$ be a finite-dimensional vector space equipped with an associative multiplication $\mu$ and a coassociative comultiplication $\delta$. Suppose that $A$, $\mu$, and $\delta$ satisfy the equations given in string notation by equations (39) and (40).

Finite-dimensional Frobenius algebra conjecture. Then $A$ admits a unit for $\mu$ and a counit for $\delta$, and is, in fact, a Frobenius algebra.

The equations are among the multiplication-comultiplication axioms of a Frobenius algebra, and the preceding discussion identifies the existence of a unit or counit as the remaining issue. The question is whether finite dimensionality forces these structures; the source presents this as an open question after unsuccessful attempts to find a counterexample.