The nearly $G_2$ conifold desingularization obstruction conjecture

Let $(\overline{X},\varphi_{CS})$ be a nearly $G_2$ conifold with a singularity at $x_0$ modelled on the $G_2$ cone $(C=C(L),\varphi_C)$. An obstruction conjecture for nearly $G_2$ conifolds. If every asymptotically conical $G_2$ manifold asymptotic to $(C,\varphi_C)$ has rate $\nu<-7/2$, then there is no smooth nearly $G_2$ manifold Gromov--Hausdorff close to $(\overline{X},\omega_{CS},\rho_{CS})$. This predicts that, under the stated asymptotic-cone condition, the nearly $G_2$ conifold cannot be approximated by smooth nearly $G_2$ manifolds; the paper presents it as a conjectural extension of the obstruction theorem to desingularizations that may occur only in a discrete sequence.