Let $\sigma(x)=1/(1+e^{-x})$ denote the sigmoid/logistic function and let $a,b,c,d>0$ such that $a\geq c$ and $a+b=c+d$. Prove that
$$ \sigma(2a-b)-\sigma(2c-d)\geq \sigma(2a+b)-\sigma(2c+d). $$
Note that both sides of the inequality are non-negative. The proof is easy if the coefficient 2 was a 1, or if the function $\sigma$ was a linear function. I simulated over 1 million values and the result seems to hold, but I'm stuck on the proof. I tried using the Lipschitz condition, subadditivity, and direct computation.