The extremal homology-manifold conjecture in dimensions 8 and 16

Let $K$ be a finite simplicial complex that is a $Z$-homology $d$-manifold, meaning that every simplex lies in a $d$-simplex and the link of each nonempty simplex $\sigma$ of dimension less than $d$ has the homology of $S^{d-dimsigma-1}$ with integer coefficients. A combinatorial manifold is a simplicial complex whose simplex links are spheres. The spaces $\mathbb{HP}^2$ and $\mathbb{OP}^2$ denote the quaternionic and octonionic projective planes.

Extremal homology-manifold conjecture.

1. Suppose that $K$ is a $15$-vertex $Z$-homology $8$-manifold such that

$$H_*(K;Z)\not\cong H_*(S^8;Z).$$

Then $K$ is a combinatorial manifold that is PL homeomorphic to $\mathbb{HP}^2$. 2. Suppose that $K$ is a $27$-vertex $Z$-homology $16$-manifold such that

$$H_*(K;Z)\not\cong H_*(S^{16};Z).$$

Then $K$ is a combinatorial manifold that is PL homeomorphic to the octonionic projective plane $\mathbb{OP}^2$.

These are extremal cases of the lower bound for non-spherical homology manifolds. The first assertion is supported by the classification and constructions in the paper, while the corresponding $16$-dimensional assertion remains open.