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Let $w$ be an equality word of two binary non-periodic morphisms $g,h:{\{a,b\}}^{*}\to {\Delta }^{*}$ 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 $w={\left(ab\right)}^{i}a$, or $w=a^i{b}^j$ with $gcd(i,j)=1$. 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 $b$.