Calabi–Yau threefolds homeomorphic to Fano threefolds

Determine whether there exists a Calabi–Yau threefold that is orientation-preservingly homeomorphic to either a smooth cubic threefold in complex projective four-space or a smooth complete intersection of two quadrics in complex projective five-space.

Background

The paper discusses whether the topology of a Calabi–Yau manifold can also support a Fano complex structure. Although strong obstructions are established in several even complex dimensions, the corresponding six-dimensional case remains unresolved. In particular, the cited problem concerns two standard Fano threefolds: a smooth cubic hypersurface in $\mathbb{C}P^4$ and a smooth intersection of two quadrics in $\mathbb{C}P^5$.