Higher-dimensional tangency bound for hypersurfaces

Let $k$ be a field and let $\mathcal{S}$ be a set of irreducible hypersurfaces in $k^d$. A directed point of tangency is a pair $(p,H)$, where $p\in k^d$ and $H$ is a hyperplane containing $p$, such that at least two distinct hypersurfaces in $\mathcal{S}$ are smooth at $p$ and tangent to $H$ at $p$. Let $\mathcal{T}(\mathcal{S})$ be the set of directed points of tangency, and let $\operatorname{mult}(p,H;\mathcal{S})$ be the number of hypersurfaces in $\mathcal{S}$ smooth at $p$ and tangent to $H$ at $p$. Let $\mathcal{S}$ be a set of $n$ irreducible hypersurfaces in $k^d$ of degree at most $D$, with $n\leq c_D\operatorname{char}(k)^d$. Higher-dimensional tangency conjecture.

$$\sum_{(p,H)\in\mathcal{T}(\mathcal{S})}\operatorname{mult}(p,H;\mathcal{S})\leq C_Dn^{(2d-1)/d}.$$

This is proposed as a higher-dimensional analogue of the paper's tangency theorem, motivated by the corresponding geometry of $(d-1)$-dimensional varieties in $\mathbb{R}^{2d-1}$. The source gives no evidence that the conjecture has been resolved.