Curvature bound and tangent-cone convergence for conifold contractions

Assume the hypotheses of the paper's main theorem, including a conifold contraction and the associated Ricci-flat Kähler metrics $g(t)$ in the classes $\alpha+t[\mathcal{L}_0]$ for $t\in(0,1]$. Let $Rm(t)$ denote the curvature tensor of $g(t)$, and let $\tilde g(t)$ be the corresponding rescaled Ricci-flat Kähler metrics. Let $\hat E$ be the exceptional locus equipped with the Ricci-flat Kähler metric $g_{CY,\hat E}$ having the isolated cone singularity described by the paper.

Curvature and rescaling conjecture. There exists $C>0$ such that, for all $t\in(0,1]$,

$$\sup_X |Rm(t)|_{g(t)}\leq Ct^{-1}.$$

Furthermore, the rescaled metrics $\tilde g(t)$ converge to $g_{CY,\hat E}$ on $\hat E$ with its isolated cone singularity in pointed Gromov–Hausdorff topology.

This conjecture concerns the curvature scale and the geometric model appearing near the exceptional locus during the degeneration. The supplied text presents it as a future question after proving the conifold-transition convergence results; no resolution is given there.