# Can it be proven that the left hand side of Robin's Inequality is bounded by some function?

Based on the shape of the curve shown in this post and the possible relationship shown here, I am willing to venture a conjecture:

$$\frac{\sigma (n)}{e^{\gamma} n \log \log n}<1-\frac{0.242692}{\ln(n)}$$

This conjecture holds for the largest Colossally Abundant Number I could find, the $$143215^{th}$$ with over 800k digits computed by Schwabhäuser. From his statistics:

$$\frac{X(n_{143215})}{e^\gamma}=0.99995934<1-\frac{0.242692}{\ln(n_{143215})}=0.999999873$$

Can this bound or something similar be proven to hold for all Superabundant Numbers (and therefore all numbers)?