The Ricci-flat TYZ-coefficient conjecture

Let $(M,g)$ be a Kähler metric on an $n$-dimensional complex manifold, and let $a_j$ denote the coefficients associated to $g$ in the Tian–Yau–Zelditch expansion. The Ricci-flat TYZ-coefficient conjecture. If $g$ is Ricci-flat and

$$a_{n+1}=0,$$

then $g$ is flat. This conjecture belongs to the study of prescribing coefficients in the TYZ expansion and extends known characterizations of flat metrics using vanishing coefficients in particular settings. Its general status is not resolved in the source.