Generalized uniqueness conjecture for compatible Calabi–Yau metrics

Fix a polystable negative valuation $v$ on $X$. Let $\mathcal M_v$ be the space of compatible Calabi–Yau metrics on $X$ with $v=v_\omega$, let $\mathcal C_v$ be the corresponding weighted asymptotic cone, and let $\mathcal N_v$ be the space of Calabi–Yau cone metrics on $\mathcal C_v$ with the Reeb vector field determined by $v$. Consider the map

$$\mathfrak C:\mathcal M_v\rightarrow\mathcal N_v$$

obtained by taking the appropriate rescaled limit of the Kähler form under the weighted asymptotic cone construction. Generalized uniqueness conjecture. The map $\mathfrak C$ is well-defined and bijective. This would identify compatible Calabi–Yau metrics with their asymptotic cone metrics and extend uniqueness beyond the uniform-equivalence question; it remains open.