The squarefree values conjecture for multivariable polynomials

Let $h\in\{1,2\}$, let $f\in\mathbb{Z}[x_1,\ldots,x_h]$ be primitive and squarefree, and let $\mathbb{Z}^h(X)=\{{\rm v}\in\mathbb{Z}^h:|{\rm v}|\leq X\}$. Define $\delta_f=\gcd\{f({\rm v}):{\rm v}\in\mathbb{Z}^h\}$, let $d_f$ be the smallest integer such that $\delta_f/d_f$ is squarefree, and let $Sqf(X)$ count the ${\rm v}\in\mathbb{Z}^h(X)$ for which $f({\rm v})d_f^{-1}$ is squarefree. With $C_f$ as defined in the source, squarefree conjecture.

$$Sqf(X)=C_fX^h+o(X^h).$$

This predicts the expected asymptotic density of squarefree values and is a key input to the paper's sieve arguments. The source presents it as an unproved conjecture.