Algebraic-variety anticoncentration conjecture for Rademacher sums

Let $0\le d<r$ and $q$ be integers. Let $\mathcal{Z}\subseteq\mathbb{R}^{r}$ be an algebraic variety of dimension $d$ and degree at most $q$. Let $\vec{a}_{1},\dots,\vec{a}_{n}\in\mathbb{R}^{r}$ be vectors such that, for some positive integer $t$, one can form $t$ disjoint bases from them. Let $(\xi_1,\dots,\xi_n)\in\{-1,1\}^{n}$ be independent Rademacher random variables. Algebraic-variety anticoncentration conjecture. There is a constant $C_{d,r,q}$ depending only on $d,r,q$ such that

$$\Pr[\xi_{1}\vec{a}_{1}+\dots+\xi_{n}\vec{a}_{n}\in\mathcal{Z}]\le\frac{C_{d,r,q}}{t^{(r-d)/2}}.$$

This conjecture would extend geometric Littlewood--Offord bounds from quadrics in affine-linear subspaces to algebraic varieties, and is related to higher-degree generalisations of the Littlewood--Offord problem. Its resolution status is not specified in the supplied text.