[Miles Reid][1] conjectured that the moduli
space of Calabi-Yau spaces $\mathscr M_{CY}$ is connected after allowing the conifold transitions
, i.e., both resolutions/contractions and deformations/degeneration. Is there any progress about this conjecture (what about in higher dimension), A reference or information is welcomed 


  [1]: https://arxiv.org/pdf/0904.4487.pdf