Positive rank gradient vanishes for residually finite groups satisfying a law

Let $\Gamma$ be a finitely generated residually finite group satisfying a non-trivial group law. The rank gradient $RG(\Gamma)$ measures the asymptotic normalized growth of the minimal number of generators of finite-index subgroups of $\Gamma$. Rank-gradient conjecture. Then

$$RG(\Gamma)=0.$$

This statement is presented as a consequence of a positive answer to A. Thom's question about whether the profinite completion of every residually finite non-amenable group contains a nonabelian free group. Its status is not resolved in the supplied text.