The exponent conjecture for Schur multipliers of finite p-groups

Let $G$ be a finite $p$-group, and write $M(G)=H_2(G,\mathbb{Z})$ for its Schur multiplier. Write $\operatorname{exp}(G)$ for the exponent of $G$. Exponent conjecture for finite $p$-groups. The exponent of the Schur multiplier divides $p$ times the exponent of the group:

$$\operatorname{exp}(M(G))\mid p\operatorname{exp}(G).$$

The paper presents this as a conjecture with no counterexample so far, motivated by counterexamples to Schur's stronger exponent conjecture for $2$-groups. Its status is not resolved in the supplied text.