Baer–Suzuki conjecture for commutators of prime-order elements

Let $G$ be a finite group and let $p$ be a prime. Let $x\in G$ be an element of order $p$. A group element is $p$-singular if its order is divisible by $p$.

Commutator Baer–Suzuki conjecture. If $[x,g]$ is either $1$ or $p$-singular for every element $g\in G$, then $x\in O_p(G)$.

The source states that no counterexample is known. This would give a further variation of the Baer–Suzuki theorem, but its resolution is not supplied.