The equitable semialgebraic regularity conjecture

Let $H$ be a $k$-partite semialgebraic hypergraph in $\mathbb{R}^d$ with total degree $D$, and let $\varepsilon>0$. An equitable partition is a partition of each vertex part into parts of equal size, with the regularity properties asserted by the equitable semialgebraic regularity lemma.

Equitable regularity conjecture. The equitable semialgebraic regularity lemma holds with partitions into

$$O_{d,k}\left((D/\varepsilon)^d\right)$$

parts.

The proved regularity lemma has a loss in the dependence on $\varepsilon$ when passing to equitable partitions. The conjecture asks whether that loss can be removed in every dimension; the source notes that this is possible by a direct argument in dimension $d=1$.