The nearly Kähler conifold desingularization obstruction conjecture

Let $(\overline{X},\omega_{CS},\rho_{CS})$ be a nearly Kähler conifold with a singularity at $x_0$ modelled on the Calabi--Yau cone $(C=C(L),\omega_C,\rho_C)$. An obstruction conjecture for nearly Kähler conifolds. If every asymptotically conical Calabi--Yau manifold asymptotic to $(C,\varphi_C)$ has rate $\nu<-3$, then there is no smooth nearly Kähler manifold Gromov--Hausdorff close to $(\overline{X},\omega_{CS},\rho_{CS})$. This is the nearly Kähler analogue of the preceding conjectural obstruction and concerns desingularizations near the conifold metric; substantial analytical challenges remain in establishing such a result in the weak holonomy setting.