A length bound for binary equality words

Let w be an equality word of two binary non-periodic morphisms g,h : {a, b}* -> Delta* with unique overflows. It is known that if w contains at least 25 occurrences of each of the letters a and b, then it has to have one of the following special forms: up to the exchange of the letters a and b either w = (ab)(i)a, or w = a(i) b(j) with gcd(j,j) = 1. We will generalize the result, justify this bound and prove that it can be lowered to nine occurrences of each of the letters a and b.