The large-prime-square-factor form of the squarefree values conjecture

Let $f\in\mathbb{Z}[x_1,\ldots,x_h]$ be primitive and squarefree, and let $\mathbb{Z}^h(X)=\{{\rm v}\in\mathbb{Z}^h:|{\rm v}|\leq X\}$. Squarefree conjecture, alternative version.

$$\#\left\{{\rm v}\in\mathbb{Z}^h(X):\text{there exists }p>X^{h/2}\text{ such that }p^2\mid f({\rm v})\right\}=o(X^h).$$

The source states that this is equivalent to the preceding squarefree conjecture and uses it to isolate the contribution of unusually large square divisors. It remains open in the setting considered.