No. Theorem 3 from "Об абелевых и центральных расширениях асферических групп" by I.S. Ashmanov and A. Yu. Olshanskii states: For any finitely based subvariety $\mathfrak{U} \subset \mathfrak{B}_n$ where $n \geq 10^{75}$ there exists a noetherian group $G$ such that $|V_\mathfrak{U}(G)| \leq n...