Optimal fractional grid bounds

Prove or disprove that the stated lower bounds for the fractional general position number of square grids are optimal for even and odd orders, respectively.

Background

The survey proposes separate lower-bound constructions for even and odd square-grid orders. It then states the conjecture that each construction gives the exact fractional general position number in its parity class.

References

The lower bound from Theorem~\ref{thm:fitzpatrick1} is optimal for even $n$ and the bound from Theorem~\ref{thm:fitzpatrick2} is optimal for odd $n$.

— The General Position Problem: A Survey  (2501.19385 - V. et al., 31 Jan 2025) in Section 7, subsection “Fractional position problems”