Punctured-disk canonical volume form conjecture

Let

$$\Delta^{*}:=\{t\in\mathbb{C}\mid 0<|t|<1\}$$

be the punctured disk, and let $h_{\mathrm{can},\Delta^{*}}$ be the canonical singular hermitian metric on $K_{\Delta^{*}}$ constructed from pluricanonical forms. Punctured-disk canonical volume form conjecture. The estimate

$$h_{\mathrm{can},\Delta^{*}}^{-1}=O\left(\frac{\sqrt{-1}\,dt\wedge d\bar t}{|t|^{2}(\log|t|)^{2}}\right)$$

holds. This concerns the growth of the invariant canonical volume form near the puncture; the author describes it as plausible and states that it is not currently solved.