The derivation construction conjecture for transposed Poisson n-Lie algebras

Let $n\geq 2$ be an integer. Let $(L,\cdot,[\cdot,\ldots,\cdot])$ be a transposed Poisson $n$-Lie algebra, and let $D$ be a derivation of both $(L,\cdot)$ and $(L,[\cdot,\ldots,\cdot])$. Define an $(n+1)$-ary operation

$$\mu(a_1,\ldots,a_{n+1})=\sum_{i=1}^{n+1}(-1)^{i-1}D(a_i)[a_1,\ldots,\hat{a}_i,\ldots,a_{n+1}],$$

where $\hat{a}_i$ denotes omission of the $i$-th entry. Derivation construction conjecture. Then $(L,\cdot,\mu(\cdot,\ldots,\cdot))$ is a transposed Poisson $(n+1)$-Lie algebra. This conjecture extends the known construction for transposed Poisson 2-Lie algebras and addresses the open question of constructing and classifying simple transposed Poisson $n$-Lie algebras. Its validity for arbitrary $n\geq 2$ and the stated derivations remains open.