The amended KSh conjecture for modular Lie algebras and Lie superalgebras

Let $p>0$. Consider simple finite dimensional Lie algebras and Lie superalgebras, including $\mathbb{Z}$-graded and non-graded examples. The KSh procedure is applied to every simple complex Lie algebra of type (1); when $p=2$, it is also applied to the relevant simple complex Lie superalgebras and their simple Volichenko subalgebras. In characteristic $2$, queerification and Jurman's examples are additionally included. From the resulting simple finite dimensional $\mathbb{Z}$-graded algebras, together with the exceptional algebras $\mathfrak{g}(A)$ for $p=5,3,2$, take non-positive parts, their complete and partial prolongations, and simple subquotients; non-graded examples are obtained by allowing deformations of the resulting algebras and Shen's variations.

Amended KSh conjecture. These constructions yield all $\mathbb{Z}$-graded simple finite dimensional examples of Lie algebras and Lie superalgebras in characteristic $p>0$, while the indicated deformations and Shen's variations yield the non-graded examples.

This is a proposed classification framework incorporating the characteristic-$2$ superization, queerification, Jurman's constructions, exceptional algebras, prolongations, subquotients, deformations, and Shen's variations. The source gives no resolution, so the conjecture remains open.