On a conjecture on transposed Poisson $ n $-Lie algebras

The notion of a transposed Poisson $ n $-Lie algebra has been developed as a natural generalization of a transposed Poisson algebra. It was conjectured that a transposed Poisson $ n $-Lie algebra with a derivation gives rise to a transposed Poisson $ (n+1) $-Lie algebra. In this paper, we focus on transposed Poisson $ n $-Lie algebras. We have obtained a rich family of identities for these algebras. As an application of these formulas, we provide a construction of $ (n+1) $-Lie algebras from transposed Poisson $ n $-Lie algebras with derivations under a certain strong condition, and we prove the conjecture in these cases.