P. Moravec's squared-exponent conjecture for the Schur multiplier

Let $G$ be a finite group. Write $\operatorname{exp}(G)$ for the exponent of $G$, and let $H_2(G,\mathbb{Z})$ be its second homology group with integer coefficients. Moravec's squared-exponent conjecture. One should have

$$\operatorname{exp}(H_2(G,\mathbb{Z}))\mid \bigl(\operatorname{exp}(G)\bigr)^2.$$

The paper presents this as a conjecture proposed before Vaughan-Lee's counterexamples to Schur's original bound, but does not report a counterexample to the squared-exponent bound itself. Its status should therefore be checked against the cited literature.