Is a $4×4$ Latin square possible whose trace-sum is 7 or 9 or 13 or 15?

Question

A Latin square with trace-sum $11$ is : $$\begin{bmatrix} 1&4&2&3\\ 2&3&4&1\\ 4&1&3&2\\ 3&2&1&4\end{bmatrix}$$ Is trace sum $7/9/5/13/15$ possible in some other arrangement?

Answer 1

By swapping numbers in the given arrangement, we can trivially derive trace sums of $7$ (main diagonal reads $2113$), $9$ ($3114$) and $13$ ($2443$).

But trace sums of $5$ and $15$ are not possible, for there would have to be three $1$s or three $4$s on the main diagonal respectively, forcing the fourth number to be the same as the other three by the Latin square property and thereby not yielding the desired trace sum.