Torsion homology growth of polynomially growing free-by-cyclic groups

We show that the homology torsion growth of a free-by-cyclic group with polynomially growing monodromy vanishes in every dimension independently of the choice of Farber chain. It follows that the integral torsion ${\rho }^{\mathbb{Z}}$ equals the ${\mathrm{\ell }}^2$-torsion ${\rho }^{(2)}$ verifying a conjecture of Lück for these groups.