Finite-plane Besicovitch set cardinality conjecture

For an odd prime power $q$, a Besicovitch set in ${\mathbb{F}_q}^2$ is a subset containing a line in every direction. Finite-plane Besicovitch set conjecture. The smallest Besicovitch set in ${\mathbb{F}_q}^2$ has cardinality

$$\frac{q(q+1)}{2}+\frac{q-1}{2}.$$

Every such set has cardinality at least $q^2/2$, while the source describes a construction attaining the displayed upper bound. The conjecture gives the exact minimum in the finite plane for odd prime powers.