Conjecture that the four distinguished combinatorial manifolds are the octonionic projective plane

Let $K_1$, $K_2$, $K_3$, and $K_4$ be the distinguished combinatorial $16$-manifolds constructed in the paper, and let $\mathbb{OP}^2$ denote the octonionic projective plane.

The distinguished-manifolds conjecture. The combinatorial manifolds $K_1$, $K_2$, $K_3$, and $K_4$ are PL homeomorphic to $\mathbb{OP}^2$.

These four manifolds are among the constructed $27$-vertex triangulations. Establishing their PL homeomorphism type would provide standard examples within the much larger family of triangulations considered in the paper; the source leaves this question open.