The dimension-three two-dimensional subalgebra conjecture for Lie $p$-algebras

Let $L$ be a Lie $p$-algebra, namely a Lie algebra equipped with a compatible $p$-operation, and suppose that $obreak\dim L>3$.

Two-dimensional subalgebra conjecture. $L$ contains a two-dimensional subalgebra.

This asks whether the algebraic-closedness hypothesis in the preceding result can be replaced by the dimension bound $\dim L>3$. The source notes that the only known counterexample to the corresponding unrestricted statement is a non-split three-dimensional simple Lie algebra.