Ideal growth in metabelian Lie $p$-algebras

Consider a finitely generated restricted Lie algebra $L$ over the finite field $\mathbb F_q$ and, given $n\ge0$, denote the number of restricted ideals $H\subset L$ with $\dim_{\mathbb F_q}L/H=n$ by $c_n(L)$. We show for the free metabelian restricted Lie algebra $L$ of finite rank that the ideal growth sequence grows superpolynomially; namely, there exist positive constants $\lambda_1$ and $\lambda_2$ such that $q^{\lambda_1n^2}\le c_n(L)\le q^{\lambda_2n^2}$ for $n$ large enough.