Conjecture on logarithmic distances between Latin squares

Let $N(n)=\{0,1,\dots,n-1\}$, and let a Latin square of order $n$ be an $n\times n$ array whose entries are symbols from $N(n)$, with each symbol occurring exactly once in every row and every column. For two Latin squares $L$ and $L'$ of order $n$, write $|L\setminus L'|$ for their Hamming distance.

Logarithmic distance conjecture. For each Latin square $L$ of order $n$,

$$\min\{|L\setminus L'|\mid L'\text{ is a Latin square of order }n\text{ and }L'\neq L\}=O(\log n).$$

The paper proves the weaker upper bound $8\sqrt n$, which is the first bound that is $o(n)$, while the conjectured logarithmic bound remains open. The back circulant Latin square provides evidence that a logarithmic order may be best possible.