Maximal root-state-complexity conjecture for binary languages

Let $T_n$ be the full transformation monoid on an $n$-element set, and let $U_{k,l}$ be the monoid defined above. For a language $L$ over an alphabet of size $2$, write $\operatorname{sc}(L)$ for its state complexity and $\operatorname{root}(L)$ for its root language. Maximal root-state-complexity conjecture. For every integer $n\geq 7$, there exist positive coprime integers $k\geq 2$ and $l\geq 3$ with $k+l=n$ such that, whenever $\operatorname{sc}(L)\leq n$,

$$\operatorname{sc}(\operatorname{root}(L))\leq |U_{k,l}|-\binom{n}{2}.$$

This bound is tight. The conjecture concerns the maximum blow-up of state complexity under the root operation; the supplied text says the construction is conjectured to achieve the maximum but gives no resolution.