The APT p-group bound for the Schur multiplier

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

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

The paper states that this conjecture was made before Vaughan-Lee produced counterexamples, but the supplied evidence does not identify which counterexample refutes this bound. The precise resolution should be checked in the cited APT reference.