The commutative 2-cocycle conjecture for classical Lie algebras in characteristic 3

Let $L$ be a finite-dimensional Lie algebra of classical type over a field of characteristic $3$, with $L e sl(2)$. A commutative $2$-cocycle is a symmetric bilinear form satisfying the commutative $2$-cocycle identity, and a symmetric bilinear invariant form is a symmetric bilinear form invariant under the Lie bracket.

Commutative 2-cocycle conjecture. The space of commutative $2$-cocycles on $L$ coincides with the space of symmetric bilinear invariant forms and is therefore $1$-dimensional, assuming the preceding one-dimensionality conjecture.

This is intended to describe commutative $2$-cocycles for classical Lie algebras in characteristic $3$, where trace forms can vanish and the usual Killing-form argument does not apply. The source presents the claim as supported by computer calculations; its status is otherwise unresolved in the supplied text.