Sarnak's volume-approximation conjecture for polynomial level sets

Let $F=(F_1,\ldots,F_p):\mathbb{R}^n\to\mathbb{R}^p$ be a polynomial map of degree $d$, let $K\subseteq\mathbb{R}^n$ be a compact set satisfying (P1), and let $T>0$ and $\alpha$ be parameters. Assume that $\\{x\in K\mid F(x)=0\\}$ is non-empty and does not lie in an $(n-p)$-dimensional linear subspace. Sarnak's conjecture. There exists a $\delta>0$ such that

\begin{aligned} \\#\Big( \\{x\in T\cdot K: \lVert F(x)\rVert\leqslant T^{d-\alpha}\\}\cap\mathbb{Z}^n\Big) &\ll \operatorname{Vol}_n\Big(\\{x\in T\cdot K: \lVert F(x)\rVert\leqslant T^{d-\alpha}\\}\Big)\\\\ &\quad+T^{\\,n-p-\delta}. \end{aligned}

This strengthens the preceding estimate by replacing its main term with the volume of the relevant sublevel set, thereby predicting that lattice-point counts are approximated by volume up to a power-saving error under the stated geometric assumptions.