Two-generator conjecture for alternating and symmetric groups

Let $k \geq 3$ and $n \geq k$. Write $\Sigma_n$ for the symmetric group on $n$ letters and $\operatorname{A}_n$ for the alternating group on $n$ letters.

Two-generator conjecture. Two elements of order $k$ suffice to generate $\Sigma_n$ when $k$ is even and to generate $\operatorname{A}_n$ when $k$ is odd.

This is presented as a conjecture concerning two-element generation of the symmetric and alternating groups by elements of prescribed order. The surrounding discussion asks for its resolution; no resolution is supplied here.