The distinct-distances conjecture for pairs of algebraic curves

Let $d\geq 2$, and let $\gamma_1,\gamma_2$ be a pair of constant-degree irreducible algebraic curves in $\mathbb{R}^d$. Assume that neither $\gamma_1$ nor $\gamma_2$ is contained in a hyperplane in $\mathbb{R}^d$. For a pair $P_1\subset\gamma_1$, $P_2\subset\gamma_2$ of $n$-point sets, the Cartesian product $P_1\times P_2$ spans distinct distances.

Distinct-distances conjecture. For every such pair of sets, $P_1\times P_2$ spans $\Omega(n^{3/2})$ distinct distances, unless each of $\gamma_1$ and $\gamma_2$ is an algebraic helix.

The conjecture seeks a characterization of the exceptional pairs of curves that can support fewer distinct distances. In the planar case, the exceptional pairs are known to be pairs of lines or concentric circles, while algebraic helices are known to play the corresponding role in the non-bipartite higher-dimensional setting.