Lewis–Sheng dimension bound for Γ-equivalent points on complete intersections

Let $X\subset {\mathbb P}^{n+k}$ be a very general complete intersection of type $(d_1,d_2,\ldots,d_k)$, let $p\in X$, and fix a smooth projective curve $\Gamma$ with distinct points $0$ and $\infty$. Define

$$R_{X,p,\Gamma}=\\{q\ne p\in X:N(p-q)\sim_\Gamma 0\text{ for some }N\in {\mathbb Z}^+\\}.$$

Lewis–Sheng dimension conjecture. The locus of points Γ-equivalent to $p$ satisfies

$$\dim R_{X,p,\Gamma}\le 2n-\sum_{i=1}^k(d_i-1).$$

This is presented as the authors’ most optimistic expectation for the optimal degree bound generalizing Voisin’s result on rationally equivalent points; its resolution is not established in the supplied text.