Multiplicity-line conjecture for small planar Besicovitch sets

Let $q$ be odd. Write $\ell(i,b_i)$ for the selected line in direction $i\in\mathbb{F}_q\cup\{\infty\}$, and let

$$B=\bigcup_{i\in\mathbb{F}_q\cup\{\infty\}}\ell(i,b_i).$$

For $P\in B$, let $m_P$ be the number of these lines passing through $P$. Call $B$ small if

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

Multiplicity-line conjecture. If $B$ is small, then there is some $j\in\mathbb{F}_q\cup\{\infty\}$ such that every point $P\in B$ with $m_P\geq3$ lies on the line $\ell(j,b_j)$. The paper proves the sharp lower bound conditionally under this structural hypothesis, but does not prove the hypothesis itself.