The toroidal no-three-in-line conjecture involving the divisor function

Let $a(n)$ be the smallest multiple of the number of divisors of $n$ that is greater than or equal to $n$. On an $n\times n$ discrete torus, points are required to have no three in a line, meaning that no three points lie in a coset of a cyclic subgroup of the torus. Toroidal no-three-in-line conjecture. Given an $n\times n$ discrete torus, one can place $a(n)$ points with no three being in a line. This conjecture proposes a general lower bound for the maximum size of a no-three-in-line configuration on a discrete torus; the paper computes several small cases but does not establish the assertion in general.