Classification conjecture for simple finite-dimensional Lie algebras with nontrivial Hom-Lie structures

Let $L$ be a simple finite-dimensional Lie algebra. A nontrivial Hom-Lie structure on $L$ is a Hom-Lie structure that is not a scalar multiple of the identity.

Classification conjecture. If $L$ admits a nontrivial Hom-Lie structure, then $L$ is isomorphic either to a $3$-dimensional simple algebra, or, in the case of positive characteristic, to the Zassenhaus algebra.

Computations for finite- and infinite-dimensional simple Lie algebras in characteristic zero motivate this classification. The conjecture concerns the exceptional simple algebras that may admit Hom-Lie structures beyond scalar multiples of the identity; its resolution is not established in the supplied text.