Conder's conjecture on simple groups with cosets of p-elements

Let $S$ be a simple group, let $p$ be a prime, and let $P$ be a Sylow $p$-subgroup of $S$. A group element is a $p$-element if its order is a power of $p$. Conder's conjecture. If a nontrivial coset $Pg$ consists entirely of $p$-elements, then $|P|=5$. The conjecture is motivated by the rarity of simple groups with this property and is supported by considerable evidence, but it remains open.