Finite field Nikodym problem for spread line sets

A set of points $N\subseteq {\mathbb{F}}_q^d$ is a Nikodym set if, for any $x\in {\mathbb{F}}_q^d$, there is a line $\mathrm{\ell }$ through $x$ such that $\mathrm{\ell }\setminus \{x\}\subseteq N$. We conjecture that $\mathrm{\mid }N\mathrm{\mid }=q^d-O_d(q^{d\mathrm{/}(d-1)})$ and prove it under an extra algebraic assumption.