Let $G$ be a finite group, and $g,h\in G$ be such that $G = \langle g,h\rangle = \langle g,hgh^{-1}\rangle$ - that is, $g,h$ generate $G$, and so does $g,hgh^{-1}$.

Let $n := |G|$, $r := |g|$, and $e := |ghg^{-1}h^{-1}|$

I was doing some computations with degenerations of covers of curves, and it seems that my computations give the following relation: $$e \ge \frac{n}{2+n-\frac{2n}{r}}$$ The cleanest special case is if $r = 2$, in which case one gets $e\ge \frac{n}{2}$ (in which case we must have $e = \frac{n}{2}$ barring the trivial case when $G$ is cyclic, so that the commutator subgroup is cyclic of index 2)

Firstly, is this true? (I've checked it to be true for the smallest 10 or so finite simple groups, where it seems to hold).

If so, this is somewhat curious to me. Is this related to any known results?