The one-dimensionality conjecture for invariant forms on classical Lie algebras

Let $L$ be a Lie algebra of classical type over a field of arbitrary characteristic. A symmetric bilinear invariant form on $L$ is a symmetric bilinear form invariant under the Lie bracket.

One-dimensionality conjecture. The space of symmetric bilinear invariant forms on $L$ is $1$-dimensional.

The claim concerns both the possibly non-simple Lie algebras obtained from the usual Chevalley basis construction in small characteristics and their simple quotients. It follows from the standard statement about the Killing form in characteristic $0$ and characteristic $p>5$, but in small characteristics the Killing and other trace forms may vanish; the source reports supporting computer calculations.