Reid's Fantasy on the connectedness of Calabi-Yau moduli

Consider the directed graph whose nodes are deformation families of smooth, compact Calabi-Yau threefolds. For two nodes $\mathcal{M}_1$ and $\mathcal{M}_2$, draw an arrow

$$\mathcal{M}_1\rightarrow\mathcal{M}_2$$

when, for a general member $X\in\mathcal{M}_1$, there is a conifold transition $\mathcal{M}_1\ni X\rightarrow X_0\leadsto X_t\in\mathcal{M}_2$. Reid's Fantasy. The graph of simply connected Calabi-Yau threefolds is connected. This formulation concerns the proposed global organization of Calabi-Yau threefold deformation families through conifold transitions; the source does not specify whether the claim has been resolved.