Compactification conjecture for polynomially asymptotically Calabi–Yau manifolds

Let $(X,I,\omega,\Omega)$ be a polynomial asymptotically Calabi Calabi-Yau manifold with rate $(\kappa_1,\kappa_2)$, meaning that it is asymptotic to a Calabi model with complex-structure and holomorphic-volume-form decay rate $\kappa_1$ and Kähler-form decay rate $\kappa_2$. There are constants $\lambda$ and $\mu$ such that

Compactification conjecture. For any $\kappa_1>\lambda$ and $\kappa_2>\mu$, $(X,I,\omega,\Omega)$ can be compactified complex analytically to a weak Fano manifold. Furthermore, the Calabi–Yau metric $\omega$ comes from the generalized Tian–Yau construction in Theorem 1.

This conjecture seeks to extend the known compactification and generalized Tian–Yau construction results from faster decay assumptions to polynomial decay. The source does not state whether the optimal constants or the asserted compactification are known, so the conjecture is treated as open.