The prime-field planar induced matching conjecture

Let $\mathrm{IM}(2,q)$ denote the maximum size of an induced matching in the point-line incidence graph of $\mathbb{F}_q^2$. Prime-field planar induced matching conjecture. There exists an absolute constant $c>0$ such that, for all sufficiently large primes $q$,

$$\mathrm{IM}(2,q)\le q^{3/2-c}.$$

For arbitrary prime powers, Vinh's finite-field Szemerédi–Trotter theorem gives $\mathrm{IM}(2,q)\le q^{3/2}+q$, and this exponent is sharp up to constants when $q$ is a square because of the Hermitian unital. The conjecture asks for a power-saving improvement over this bound over prime fields.