Herzog–Kaplan–Lev product-length conjecture for cycle classes

Let $n(k,l)$ be the largest integer $n$ such that every element of the alternating group $A_n$ is a product of $k$ many $l$-cycles, where $k,l\geq 2$ and either $l$ is odd or $k$ is even. Herzog–Kaplan–Lev conjecture.

$$\left\lfloor\frac{2}{3}kl\right\rfloor\leq n(k,l)\leq \left\lfloor\frac{2}{3}kl\right\rfloor+1.$$

This conjecture generalizes known results on products of cycle classes; it was verified in several cases, including $3\mid l$, $l\geq 9$, and odd $l$, but remains open in general.