Abelianness of group gradings that are coarsenings of abelian gradings

Let $L$ be a Lie algebra with a group grading $\Gamma$ that is a coarsening of an abelian grading of $L$. Abelianness conjecture. Then $\Gamma$ is abelian. The paper states that no proof is known; the claim concerns whether coarsening an abelian grading can produce a nonabelian group grading.