The extract criterion for irreducibility of holomorphic eta quotients

Call an eta quotient primitive if it is not a rescaling of another eta quotient by a positive integer. For an eta quotient $f$, its extract is the primitive eta quotient $f_0$ of which $f$ is a rescaling by a positive integer. Extract criterion. A holomorphic eta quotient is irreducible if its extract is quasi-irreducible. The source presents this as a restatement implied by the rescaling irreducibility conjecture; no independent resolution is supplied.