On the asymptotic Makar-Limanov rank conjecture

Let $\mathbb{k}$ be an algebraically closed field of characteristic zero and let $f$ be a nonconstant polynomial in finitely many freely noncommuting variables. We prove the asymptotic Makar-Limanov rank conjecture, namely that the normalized rank of a value of $f$ can be made arbitrarily small by choosing matrices over $\mathbb{k}$ of finite size.