By Theorem 1.2 in this paper, Robin's Inequality is true for every odd integer $n>10$. 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 second 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.

By Theorem 1.2 in this paper, Robin's Inequality is true for every odd integer $n>10$. 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 second next one (cf. Proposition 4 in this paper). So, by Proposition 1 in Section 3 of Robin's paper, we would 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.

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.

By Theorem 1.2 in this paper, Robin's Inequality is true for every odd integer $n>10$. 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 second 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.