A refined energy bound for perpendicular bisectors

Let $\mathcal{P}$ be a set of $n$ points in the Euclidean plane. We prove that, for any $\epsilon > 0$, either a single line or circle contains $n/2$ points of $\mathcal{P}$, or the number of distinct perpendicular bisectors determined by pairs of points in $\mathcal{P}$ is $\Omega(n^{52/35 - \epsilon})$, where the constant implied by the $\Omega$ notation depends on $\mathcal{P}$. This is progress toward a conjecture of Lund, Sheffer, and de Zeeuw, that either a single line or circle contains $n/2$ points of $\mathcal{P}$, or the number of distinct perpendicular bisectors is $\Omega(n^2)$. The proof relies bounding the size of a carefully selected subset of the quadruples $(a,b,c,d) \in \mathcal{P}^4$ such that the perpendicular bisector of $a$ and $b$ is the same as the perpendicular bisector of $c$ and $d$.