Chen–Lewis–Sheng conjecture on rationally equivalent points on complete intersections

Let $X \subset \mathbb{P}^n$ be a very general complete intersection of multidegree $(d_1, \dots, d_k)$. For $p \in X$, let $R_{X,p}$ denote the space of points of $X$ rationally equivalent to $p$. Chen–Lewis–Sheng conjecture. For every $p \in X$, $R_{X,p}$ has dimension at most

$$2n-k-\sum_{i=1}^k d_i.$$

If $2n-k-\sum_{i=1}^k d_i<0$, this means that $p$ is equivalent to no other point of $X$. The paper states that it proves the remaining cases of this conjecture, so the conjecture is solved by the results discussed here.