Carpi's conjecture on abelian power-free morphisms

Let $\Sigma$ be a finite alphabet, and let a morphism on $\Sigma$ be called abelian power-free when its images avoid abelian powers of the relevant exponent. Carpi's set of conditions is the criterion for abelian power-freeness referred to in the source. Carpi's conjecture. For alphabets of size less than six, Carpi's conditions are necessary and sufficient for a morphism to be abelian power-free. Carpi's conditions are already known to give a complete characterization for alphabets of size at least six, while necessity and sufficiency in the smaller-alphabet case are posed here as unresolved.