The right-angle 2-chain conjecture in three dimensions

Let $E\subset\mathbb R^3$ be a large finite point set of $n$ points. An angle 2-chain consists of three consecutive angles determined by four points, and $\Lambda_2(E)$ denotes the set of such chains; here both angles are right angles. Right-angle 2-chain conjecture. For right angles,

$$|\Lambda_2(E)|\lesssim n^3.$$

This would improve the currently available trivial upper bound of $\lesssim n^{10/3}$ for 2-chains in three dimensions. The conjecture is presented as a wide open problem; no nontrivial bounds are known for chains of non-right angles in three dimensions.