Via integer linear programming, I found the following minimum values for $n \times n$ grids:
>! \begin{matrix} n & \min \\ \hline 3 & 4 \\ 4 & 6 \\ 5 & 8 \\ 6 & 11 \\ 7 & 14 \\ 8 & 16 \\ 9 & 20 \\ 10 & 24 \\ 11 & 28 \\ 12 & 32 \end{matrix}

Minimum values for $n \le 44$ are here: https://oeis.org/A365271

8x8:
>! [![enter image description here][1]][1]

9x9:
>! [![enter image description here][2]][2]

10x10:
>! [![enter image description here][3]][3]

11x11:
>! [![enter image description here][4]][4]

12x12:
>! [![enter image description here][5]][5]


  [1]: https://i.sstatic.net/t8lV5vyf.png
  [2]: https://i.sstatic.net/O9wloSb1.png
  [3]: https://i.sstatic.net/8MBrbn1T.png
  [4]: https://i.sstatic.net/A0ndvb8J.png
  [5]: https://i.sstatic.net/UcdCf1ED.png