Prove the non-isomorphism conjecture for nonzero semisimple Lie algebras

Prove that, for every nonzero semisimple Lie algebra g, the spectral-cocycle vertex algebroid extension A_lambda(A,g[u]) is not isomorphic to any level-and-closed-three-form extension A_{\widehat\kappa,\mathcal H}(A,g[u]) for any invariant level \widehat\kappa and closed three-form \mathcal H.

Background

The paper constructs a spectral-cocycle extension of the level-zero Yangian vertex algebroid using a nontrivial Chevalley–Eilenberg cocycle associated with spectral translation. For reductive Lie algebras with nontrivial center, Theorem [thm:spectral-ext-param] proves that this extension cannot be obtained, up to isomorphism, from the level-and-closed-three-form family.

The authors then propose that the same distinction persists for nonzero semisimple Lie algebras, where the center vanishes and the argument used for reductive Lie algebras with nontrivial center does not apply directly. Establishing the conjecture would show that spectral-cocycle extensions provide genuinely new isomorphism classes beyond the level-and-three-form constructions in the semisimple case.