Faber's lower-bound conjecture for Kakeya sets in odd-order projective planes

Let $\pi_q$ be a projective plane of odd order $q$, and let $K$ be a Kakeya set in $\pi_q$, meaning that $K$ contains at least one point on each line of a fixed line pencil. Faber's conjecture. Every Kakeya set $K$ satisfies

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

For even $q$, the corresponding lower bound $|K|\ge q(q+1)/2$ is attained by the construction from a dual hyperoval; when $q$ is odd, the conjectured bound is motivated by the absence of hyperovals and by the associated construction from a dual oval. The conjecture was raised and studied by X. W. C. Faber; the supplied text does not establish whether it has been resolved.