Exponent-square conjecture for the Schur multiplier

Let $G$ be a finite group, and let $H_2(G,\mathbb{Z})$ denote its Schur multiplier. Exponent-square conjecture.

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

This conjecture was proposed as a proposed replacement for Schur's exponent conjecture after counterexamples to the latter were found. The supplied parser marks it as refuted, although the surrounding excerpt only explicitly says that the cited counterexamples to Schur's conjecture are not counterexamples to this statement.