The algebraic Erdős–Ko–Rado conjecture for finite 2-transitive groups

Let $G$ be a finite $2$-transitive group. For each coset of a point stabiliser in $G$, let its characteristic vector be the corresponding element of the group algebra $\mathbb{C}G$, and let $V$ be the subspace spanned by these characteristic vectors. If $S$ is an intersecting set of maximal cardinality in $G$, write $\chi_S$ for its characteristic vector. Algebraic Erdős–Ko–Rado conjecture. Then

$$\chi_S\in V.$$

This conjecture proposes an algebraic analogue of the Erdős–Ko–Rado theorem. The paper proves the assertion for some $2$-transitive groups, but its validity for arbitrary finite $2$-transitive groups remains open.