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.