Coincidence of fine gradings and fine group gradings on simple complex Lie algebras

Let $L$ be a simple complex Lie algebra. A grading of $L$ is called fine if it cannot be properly refined by another grading, and a fine group grading is a fine grading whose index set carries a group structure compatible with the grading. Fine-grading coincidence assumption. On a simple complex Lie algebra, the terms fine grading and fine group grading coincide. This would extend the authors' classification of fine group gradings to all fine gradings for the simple complex Lie algebras considered in their work; the statement is presented as an assumption and its resolution is not given here.