Nikodym set size conjecture for finite fields

Let $d\geq 3$, let $\mathbb{F}_q^d$ be the $d$-dimensional vector space over the finite field with $q$ elements, and let a weak Nikodym set be a subset of $\mathbb{F}_q^d$ containing, for every point $x\in\mathbb{F}_q^d$, all but possibly $x$ of the points on some line through $x$. Nikodym set size conjecture. For any $d\geq 3$, there exists a constant $C_d>0$ such that every weak Nikodym set $N\subseteq\mathbb{F}_q^d$ satisfies

$$|N|\geq q^d-C_dq^{d-\frac{1}{d}}.$$

The conjecture strengthens the known lower bounds for finite-field Nikodym sets and is motivated by the fact that all known constructions have size $(1-o_{d;q\to\infty}(1))q^d$.