Euler's Conjecture on Orthogonal Latin Squares

Conjecture
Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Two orthogonal Latin squares do not exist for order $4 n + 2$.

Refutation
While the conjecture is true for order $6$, it fails for order $10, 14, \ldots$.

The following is an example of two orthogonal Latin squares presented in the same array.

The elements of the first Latin square are shown in normal type, while those of the second Latin square are shown in italic type:

$\begin{array} {|c|c|c|c|c|c|c|c|c|c|} \hline 4 \mathit 6 & 5 \mathit 7 & 6 \mathit 8 & 7 \mathit 0 & 8 \mathit 1 & 0 \mathit 2 & 1 \mathit 3 & 2 \mathit 4 & 3 \mathit 5 & 9 \mathit 9 \\ \hline 7 \mathit 1 & 9 \mathit 4 & 3 \mathit 7 & 6 \mathit 5 & 1 \mathit 2 & 4 \mathit 0 & 2 \mathit 9 & 0 \mathit 6 & 8 \mathit 8 & 5 \mathit 3 \\ \hline 9 \mathit 3 & 2 \mathit 6 & 5 \mathit 4 & 0 \mathit 1 & 3 \mathit 8 & 1 \mathit 9 & 8 \mathit 5 & 7 \mathit 7 & 6 \mathit 0 & 4 \mathit 2 \\ \hline 1 \mathit 5 & 4 \mathit 3 & 8 \mathit 0 & 2 \mathit 7 & 0 \mathit 9 & 7 \mathit 4 & 6 \mathit 6 & 5 \mathit 8 & 9 \mathit 2 & 3 \mathit 1 \\ \hline 3 \mathit 2 & 7 \mathit 8 & 1 \mathit 6 & 8 \mathit 9 & 6 \mathit 3 & 5 \mathit 5 & 4 \mathit 7 & 9 \mathit 1 & 0 \mathit 4 & 2 \mathit 0 \\ \hline 6 \mathit 7 & 0 \mathit 5 & 7 \mathit 9 & 5 \mathit 2 & 4 \mathit 4 & 3 \mathit 6 & 9 \mathit 0 & 8 \mathit 3 & 2 \mathit 1 & 1 \mathit 8 \\ \hline 8 \mathit 4 & 6 \mathit 9 & 4 \mathit 1 & 3 \mathit 3 & 2 \mathit 5 & 9 \mathit 8 & 7 \mathit 2 & 1 \mathit 0 & 5 \mathit 6 & 0 \mathit 7 \\ \hline 5 \mathit 9 & 3 \mathit 0 & 2 \mathit 2 & 1 \mathit 4 & 9 \mathit 7 & 6 \mathit 1 & 0 \mathit 8 & 4 \mathit 5 & 7 \mathit 3 & 8 \mathit 6 \\ \hline 2 \mathit 8 & 1 \mathit 1 & 0 \mathit 3 & 9 \mathit 6 & 5 \mathit 0 & 8 \mathit 7 & 3 \mathit 4 & 6 \mathit 2 & 4 \mathit 9 & 7 \mathit 5 \\ \hline 0 \mathit 0 & 8 \mathit 2 & 9 \mathit 5 & 4 \mathit 8 & 7 \mathit 6 & 2 \mathit 3 & 5 \mathit 1 & 3 \mathit 9 & 1 \mathit 7 & 6 \mathit 4 \\ \hline \end{array}$

Note that all integers from $0 \mathit 0$ to $9 \mathit 9$ appear exactly once each.