Tangent-tube counting conjecture

Let $B_R\subset{\mathbb R}^n$ be a ball of radius $R$, and let $\Omega$ be a collection of $(R^{1/2},R)$-tubes in $B_R$ with $R^{-1/2}$-separated directions. Suppose the tubes are tangent to an $(n-1)$-dimensional variety $Z$ of degree at most $D$. Tangent-tube counting conjecture. There is a constant $C$ independent of $D$ and $R$ such that

$$\#\Omega{\;\lessapprox\;}D^C R^{\frac{n-2}{2}}.$$

This conjecture is known when $n=3$ and has a weaker form in dimension four, where $D^C$ is replaced by an unspecified constant depending on $D$. Its validity in higher dimensions is unknown.