Holzer and König's maximal two-generated transformation monoid conjecture

For $n e 7$ be an integer, choose coprime integers $k,l e 2$ with $k+l=n$, and let $U_{k,l}$ be the transformation monoid defined by the two-cycle permutation construction above. Holzer and König's conjecture. For every $n e 7$, there exist coprime integers $k,l e 2$ with $k+l=n$ such that $U_{k,l}$ is the largest two-generated submonoid of $T_n$. The claim extends the corresponding theorem for \prime $n$ to all $n$; its status is not established in the supplied text.