Kervaire's classification conjecture for 4-connected 10-manifolds

Let $M$ be a $4$-connected closed manifold of dimension $10$. Its $5$-th Betti number is denoted by $B_5(M)$, and its Kervaire invariant is $\Phi(M)\in \mathbb{Z}_2$.

Kervaire's classification conjecture. The $5$-th Betti number $B_5(M)$ and Kervaire invariant $\Phi(M)\in \mathbb{Z}_2$ are a complete set of invariants of the homotopy type of $M$.

This conjecture concerns the classification of $4$-connected closed $10$-manifolds by homotopy type and arises from the question of whether vanishing Kervaire invariant guarantees the existence of a smooth structure. Its resolution status is not specified in the source.