Free topological universal algebras and absolute neighborhood retracts

We prove that for a complete quasivariety $\mathcal K$ of topological $E$-algebras of countable discrete signature $E$ and each submetrizable $\mathsf{ANR}(k_\omega)$-space $X$ its free topological $E$-algebra $F_\mathcal K(X)$ in the class $\mathcal K$ is a submetrizable $\mathsf{ANR}(k_\omega)$-space.