The Poisson n-Lie algebra construction conjecture

Let $(\mathcal A, \cdot)$ be a unital commutative associative algebra equipped with mutually commuting derivations $d_1, \dots, d_{n+m}$ satisfying Assumptions 1 and 2, and let $A$ be a scalar matrix of size $(n+m) \times m$.

Poisson $n$-Lie algebra construction conjecture. Then $(\mathcal A, \cdot, [-,\dots,-])$ forms a Poisson $n$-Lie algebra.

This conjecture extends the construction verified in the preceding cases $n=3,4$ and asserts that the resulting bracket satisfies the Poisson $n$-Lie algebra identities in general. The status of the conjecture is not specified in the source.