The higher-wise independence conjecture for projective norm graphs

Let $q$ be a prime power, let $4\leq\ell<t$ be integers, and let $\operatorname{NG}(q,t)$ denote the projective norm graph. A set of vertices has common neighbours if those vertices are all adjacent to each such common vertex.

Higher-wise independence conjecture. For any prime power $q$ and integers $4\leq\ell<t$, all but $o(n^\ell)$ sets of $\ell$ vertices in $\operatorname{NG}(q,t)$ have $(1+o(1))q^{t-\ell}$ common neighbours.

The claim extends the established asymptotic description of common neighbourhoods from triples to larger sets of vertices. The paper presents it as being supported by computer calculations; its general validity remains open.