Maximally nonassociative quasigroups via quadratic orthomorphisms

A quasigroup $Q$ is said to be \emph{maximally nonassociative} if $x\cdot (y\cdot z) = (x\cdot y)\cdot z$ implies $x=y=z$, for all $x,y,z\in Q$. We show that, with finitely many exceptions, there exists a maximally nonassociative quasigroup of order $n$ whenever $n$ is not of the form $n=2p_1$ or $n=2p_1p_2$ for primes $p_1,p_2$ with $p_1\le p_2<2p_1$.