The no-heavy-curves conjecture for few-distance point sets

Let $\mathcal P$ be a set of $n$ points in $\mathbb R^2$, and let $D(\mathcal P)$ denote the number of distinct distances determined by $\mathcal P$. No-heavy-curves conjecture. If

$$D(\mathcal P)=o(n),$$

then there does not exist a constant-degree curve containing $\Omega(n^{1/2+\varepsilon})$ points of $\mathcal P$, for any $\varepsilon>0$. This would extend the paper's separate bounds for lines, circles, and constant-degree algebraic curves containing no line or circle; the supplied text gives no resolution, so the conjecture remains open.