Prime-modulus rank-count conjecture for cubic forms

Let $C(\mathbf{x})=\sum_{i,j,k}c_{ijk}x_ix_jx_k$ be a cubic form in $n$ variables with symmetric integer coefficients, and let $M(\mathbf{x})$ be the matrix with entries $M(\mathbf{x})_{jk}=\sum_{i=1}^n c_{ijk}x_i$. For a prime $p$, write $r_p(\mathbf{x})$ for the rank of $M(\mathbf{x})$ over $\mathbb{F}_p$. Assume that $C$ satisfies Davenport's Geometric Condition:

$$\#\{\mathbf{x}\in\mathbb{Z}^n:\|\mathbf{x}\|_\infty\leq P,\ \operatorname{rank}M(\mathbf{x})=r\}\ll P^{r+\varepsilon}$$

for every integer $r$ with $0\leq r\leq n$.

Prime-modulus rank-count conjecture. For all $n$, uniformly in $1\leq H\leq R$, and for every relevant rank threshold $r$,

$$\#\{\mathbf{h}\leq H,\ R<p\leq2R:\ r_p(\mathbf{h})\leq r\}\ll H^rR^{1+\varepsilon}.$$

This conjecture concerns the distribution of low-rank reductions modulo primes and is proposed to control the remaining possible obstructions to absolute convergence of the singular series in the nine-variable case.