A note on nonlinear mixed bi-skew Jordan and bi-skew Lie $n$-derivations on $\ast$-algebras

Let $\mathfrak{A}$ be a unital $\ast$-algebra with identity $\mathcal{I}$. For $\mathcal{A},\mathcal{B}\in \mathfrak{A}$, define the bi-skew Jordan product

$$\mathcal{A}\bullet \mathcal{B}=\mathcal{A}{\mathcal{B}}^{\ast }+\mathcal{B}{\mathcal{A}}^{\ast }$$

and the bi-skew Lie product

$$[\mathcal{A},\mathcal{B}{]}_{\diamond }=\mathcal{A}{\mathcal{B}}^{\ast }-\mathcal{B}{\mathcal{A}}^{\ast }.$$

Suppose that a nonlinear mapping $\mathrm{\Phi }:\mathfrak{A}\to \mathfrak{A}$ satisfies

$$\mathrm{\Phi }\left([{\mathcal{A}}_1\bullet {\mathcal{A}}_2\bullet \cdots \bullet {\mathcal{A}}_{n-1},{\mathcal{A}}_n{]}_{\diamond }\right)=\sum _{j=1}^n[{\mathcal{A}}_1\bullet \cdots \bullet {\mathcal{A}}_{j-1}\bullet \mathrm{\Phi }({\mathcal{A}}_j)\bullet {\mathcal{A}}_{j+1}\bullet \cdots \bullet {\mathcal{A}}_{n-1},{\mathcal{A}}_n{]}_{\diamond }$$

for all suitable elements ${\mathcal{A}}_1,\dots ,{\mathcal{A}}_n\in \mathfrak{A}$, where ${\mathcal{A}}_1,{\mathcal{A}}_2\in \{\mathcal{I},i\mathcal{I}\}$ and ${\mathcal{A}}_j=\mathcal{I}$ for every $j=3,4,\dots ,n-2$. We prove that $\mathrm{\Phi }$ is an additive $\ast$-derivation on $\mathfrak{A}$. As applications, several consequences are obtained for prime $\ast$-algebras, factor von Neumann algebras, von Neumann algebras without central summands of type ${\mathcal{I}}_1$, and standard operator algebras. Moreover, a conjecture is proposed to motivate further research in this direction. The obtained results extend and generalize a recent result of Abbasi et al. [Non-additive mixed bi-skew Jordan and bi-skew Lie triple derivations on $\ast$-algebras, Ricerche Mat., 2026.] concerning non-additive mixed bi-skew Jordan and bi-skew Lie triple derivations on $\ast$-algebras.