Sharp lower-bound conjecture for odd-order planar Besicovitch sets

Let $q$ be odd, and write $\ell(m,b)$ and $\ell(\infty,a)$ for the lines $y=mx+b$ and $x=a$ in $\mathbb{F}_q^2$. A Besicovitch set is a set $B\subset\mathbb{F}_q^2$ containing one line in every direction. For $P\in B$, let $m_P$ be the number of selected lines passing through $P$. Sharp lower-bound conjecture. Every Besicovitch set satisfies

$$\sum_{P\in B}\frac{(m_P-1)(m_P-2)}{2}\geq\frac{q-1}{2},$$

and hence

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

The bound is motivated by the explicit example discussed in the paper, while the unconditional theorem only proves the weaker estimate $q/3$ (with a later refinement to $(5q-1)/14$), so the conjecture remains open in the source.