Suppose that two orthogonal $4x4$ Latin Squares both have $1,2,3,4$ as the main diagonal. Is it possible for both of them to have the same $(2,3)$ entry?
My thinking was to write out Latin squares of order $4$ but there are $576$ of them....
So I am not quite sure how to proceed with this,
Is the main diagonal considered this
$$ \begin{bmatrix}1 & & & \\&2&&\\ &&3&\\&&&4\end{bmatrix}$$
Or is it considered this
$$ \begin{bmatrix} & & & 1\\&&2&\\ &3&&\\4&&&\end{bmatrix}$$
And then is it possible for them to have the same $(2,3)$ entry? If so what is the latin square?
Usually the main diagonal would be your first diagram.
As I interpret your question, the order along the main diagonal is also matched between the two squares, in which case the answer to the opening question is naturally "no", since paired entries $(1\;1),$ $(2\;2),$ $(3\;3),$ $(4\;4)$ already exist and another matching pair would thus break orthogonality.