Erdős's conjecture on equilateral triangles in six dimensions

Let $T_6(n)$ denote the maximum number of equilateral triangles determined by a set of $n$ points in $\mathbb{R}^6$. Erdős's conjecture. Any set of $n$ points in $\mathbb{R}^6$ can span at most

$$n^3/27+o(n^3)$$

equilateral triangles, i.e., $T_6(n)\leq n^3/27+o(n^3)$. Erdős and Purdy proved the matching lower bound $T_6(n)\geq n^3/27-O(n^2)$, so the conjecture would determine the asymptotic maximum. Its resolution remains open.