Locally conformal parallel $G_2$ and $Spin(7)$ manifolds
  We characterize compact locally conformal parallel $G_2$ (respectively, $Spin(7)$) manifolds as fiber bundles over $S^1$ with compact nearly K\"ahler (respectively, compact nearly parallel $G_2$) fiber. A more specific characterization is provided when the local parallel structures are flat. 
Stefan Ivanov, Maurizio Parton, Paolo Piccinni