Discrete bilinear-form values conjecture

Let $F$ be the real or complex field, let $\omega$ be a non-degenerate symmetric or skew-symmetric bilinear form on $F^2$, and let $P\subset F^2\setminus\{0\}$ be an $N$-element point set. Define

$$T_\omega(P)=\{\omega(q,q'):q,q'\in P\}\setminus\{0\}.$$

The exceptional case in which $T_\omega(P)$ is empty occurs when $\omega$ is skew-symmetric and $P$ is supported on a single line through the origin. Discrete bilinear-form values conjecture. Outside this exceptional case,

$$|T_\omega(P)|=\Omega^*(N),$$

meaning a lower bound of order $N$ up to possible logarithmic factors in $N$. This is presented as a central open question in discrete projective plane geometry; the paper proves the weaker bound $\Omega(N^{9/13})$, while the conjectured near-linear estimate remains open.