Squarefree-value conjecture for integral multivariate polynomials

Let $P\in\mathbb{Z}[X_{1},\ldots,X_{n}]$ have total degree $d\geq 2$. For $m>1$, define

$$\rho_{P}(m)=\#\{X\in(\mathbb{Z}/m\mathbb{Z})^{n}:P(X)\equiv0\pmod m\}.$$

Given $B_j\in\mathbb{R}$ with $B_j\geq1$ and $B=\prod_{j=1}^{n}[0,B_j]\cap\mathbb{Z}^{n}$, let $N_P(B)$ count the $X\in B$ for which $P(X)$ is nonzero and square-free. Squarefree-value conjecture.

$$N_{P}(B)\sim \mathscr{C}_{P}B_{1}\cdots B_{n}\quad\text{as }\min_{j=1,\ldots,n}B_j\to\infty,$$

where

$$\mathscr{C}_{P}=\prod_{p}\left(1-\frac{\rho_{P}(p^{2})}{p^{2n}}\right).$$

This is the conjectural squarefree density for values of multivariate integral polynomials, with the Euler product giving the predicted local density. The supplied text does not specify whether it has been resolved in the stated generality.