Herzog–Kaplan–Lev conjecture on products of cycle classes

Let $k\geq 2$ and $l\geq 3$, with either $l$ odd or $k$ even. For $2\leq l\leq n$, let $C_l$ be the conjugacy class of $l$-cycles in $S_n$, and let $n(k,l)$ be the largest natural number $n$ such that $C_l^k=A_n$.

Herzog–Kaplan–Lev conjecture. One has

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

The conjecture is false in general: the paper determines $n(k,l)$ for $k\geq 5$ and $3\nmid l$, obtaining values below the conjectured lower bound. It is nevertheless true in several cases, including $k=2,3,4$ and $3\mid l$.