Fitzpatrick's fractional general position conjecture for square grids

Let $P_n$ be the path on $n$ vertices, let $P_n\mathbin{\square}P_n$ be the $n\times n$ Cartesian grid, and let $\operatorname{gp}_f$ denote the fractional general position number. For even $n\geq 3$, define the lower bound from Theorem 2.1 by

$$\frac{n^2-2t(t+1)}{n-t-\frac12}$$

for $0\leq t\leq n/2$. For odd $n\geq 3$, use the two lower bounds from Theorem 2.2, namely

$$\frac{n^2+1-8k^2}{n-2k}$$

and

$$\frac{n^2-1-8(k^2+k)}{n-2k-1}$$

with the parameter ranges specified there. Fitzpatrick's conjecture. The lower bound from Theorem 2.1 is optimal for even $n$, and the bound from Theorem 2.2 is optimal for odd $n$.

Exact values were known in the survey only for small orders, although the lower bound had been shown best possible through $n=10$; the conjectured optimality for all even and odd orders remains open.