Codimensions of polynomial identities of representations of Lie algebras

Consider a representation $\rho \text{\hspace{0.17em}⁣}:L\to \mathfrak{g}\mathfrak{l}(V)$ where $L$ is a Lie algebra and $V$ is a finite dimensional vector space. We prove the analog of Amitsur’s conjecture on asymptotic behavior for codimensions of polynomial identities of $\rho$.