"Motto" Moduli space of Calabi-Yau varieties can be connected by using Symplectic surgery theory.

Such surgery are very important for study of Hermitian–Yang–Mills metrics under conifold transitions

A conifold transition is a surgery on a (real) six-dimensional manifold $X$ which replaces a three-sphere with trivial normal bundle by a two-sphere with trivial normal bundle, cutting out $S^3 × D^3$ and replacing it with $D^4 × S^2$.

Miles Reid 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 . See some partial result