Asymptotic growth conjecture for right-angle-free subsets of finite vector spaces

Let $R(n,q)$ denote the maximum cardinality of a subset of the vector space $4\mathbb{F}_q^n4$ containing no right angles. The notation $4\Theta4$ refers to asymptotic equality up to positive constant factors as $n$ varies with $q$ fixed. Asymptotic growth conjecture. For every prime power $q$,

$$R(n,q)=\Theta(n^{q-1}).$$

The paper proves matching polynomial-order upper and lower bounds when $q$ is odd, while the assertion for even prime powers remains open.