Let q be an odd prime power and suppose that
$a,b\in \mathbb {F}_q$ are such that
$ab$ and
$(1{-}a)(1{-}b)$ are nonzero squares. Let
$Q_{a,b} = (\mathbb {F}_q,*)$ be the quasigroup in which the operation is defined by
$u*v=u+a(v{-}u)$ if
$v-u$ is a square, and
$u*v=u+b(v{-}u)$ if
$v-u$ is a nonsquare. This quasigroup is called maximally nonassociative if it satisfies
$x*(y*z) = (x*y)*z \Leftrightarrow x=y=z$. Denote by
$\sigma (q)$ the number of
$(a,b)$ for which
$Q_{a,b}$ is maximally nonassociative. We show that there exist constants
$\alpha \approx 0.029\,08$ and
$\beta \approx 0.012\,59$ such that if
$q\equiv 1 \bmod 4$, then
$\lim \sigma (q)/q^2 = \alpha $, and if
$q \equiv 3 \bmod 4$, then
$\lim \sigma (q)/q^2 = \beta $.