On the conjecture of Kashuba and Mathieu about free Jordan algebras

Kashuba and Mathieu proposed a conjecture on vanishing of Lie algebra homology, implying a description of the $G{L}_d$-module structure of the free $d$-generated Jordan algebra. Their conjecture relies on a functorial version of the Tits–Kantor–Koecher construction that builds Lie algebras out of Jordan algebras. In this note, we summarize new intricate computational data concerning free Jordan algebras and explain why, despite a lot of overwhelmingly positive evidence, the conjecture of Kashuba and Mathieu is not true.