The rational (non-)formality of the non-3-equal manifolds

Let $M_d^{(k)}(n)$ be the manifold of $n$-tuples $(x_1,\dots ,x_n)\in ({\mathbb{R}}^d{)}^n$ having non-$k$-equal coordinates. We show that, for $d\ge 2$, $M_d^{(3)}(n)$ is rationally formal if and only if $n\le 6$. This stands in sharp contrast with the fact that all classical configuration spaces $M_d^{(2)}(n)=\text{Conf}({\mathbb{R}}^d,n)$ are rationally formal, just as are all complements of arrangements of arbitrary complex subspaces with geometric lattice of intersections. The rational non formality of $M_d^{(3)}(n)$ for $n>6$ is established via detection of non-trivial triple Massey products assessed through Poincaré duality.