By Theorem 1.2 in this paper, Robin's Inequality is true for every odd integer $n>8$. If we knew what the OP wants to prove, then we would also know Robin's Inequality for every integer $n$ whose odd part exceeds $5040$. In particular, we would know Robin's Inequality for every colossally abundant number exceeding $5040$, because each colossally abundant number divides the next one. So, by Proposition 1 in Section 3 of Robin's paper, we would even know Robin's Inequality for every integer exceeding $5040$, which is equivalent to the Riemann Hypothesis.
In short, it is hopeless to prove what the OP wants to prove, because it implies the Riemann Hypothesis.