Lattice-minimality conjecture for asymptotic distinct-triangle counts

Let $c_0$ be the real number such that the $n$-point square lattice spans

$$c_0 n^2+o(n^2)$$

distinct triangles. For distinct nonzero vectors $a,b\in\mathbb R^2$, define the $n$-point lattice generated by $a$ and $b$ by

$$\left\{ua+vb\mid 0\leq u,v<\sqrt n,\ u,v\in\mathbb Z\right\}.$$

Lattice-minimality conjecture. This lattice spans at least

$$c_0 n^2+o(n^2)$$

distinct triangles. The conjecture asserts that no two-dimensional lattice has an asymptotically smaller distinct-triangle count than the square lattice; the source gives no resolution.