By Theorem 1.2 in [this paper][1], 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][2] exceeding $5040$, because each colossally abundant number divides the next one. So, by Proposition 1 in Section 3 of [Robin's paper][3], 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.

  [1]: https://jtnb.centre-mersenne.org/item/?id=JTNB_2007__19_2_357_0
  [2]: https://en.wikipedia.org/wiki/Colossally_abundant_number
  [3]: http://zakuski.utsa.edu/~jagy/Robin_1984.pdf