Classification by the Bianchi–Massey and pentagonal Massey tensors

Let $X$ and $Y$ be closed $(n{-}1)$-connected $m$-manifolds, with $n>1$ and $m\leq 5n-2$. Let

$$F:H^*(X)\to H^*(Y)$$

be an isomorphism of cohomology algebras. Write $\mathcal{T}_b$ and $\mathcal{T}_c$ for the uniform triple products associated with choices $b$ on $X$ and $c$ on $Y$, write $\mathcal{P}_b$ and $\mathcal{P}_c$ for the corresponding pentagonal Massey tensors, and let $\Delta$ be the subspace of $\operatorname{Hom}(\mathcal{D}^*,H^{*-2})$ defined from these choices. Classification conjecture. The map $F$ is realised by a rational homotopy equivalence if and only if $F$ intertwines the Bianchi–Massey tensors and, for every pair of choices $b$ and $c$ such that $F$ intertwines $\mathcal{T}_b$ and $\mathcal{T}_c$, one has

$$F^\#\mathcal{P}_c-\mathcal{P}_b\in\Delta.$$

The conjecture aims to extend the known classification result below dimension $5n-2$ by identifying the additional fourfold-product information needed in the borderline range; the surrounding discussion notes that data beyond these tensors may be required above this range.