Is this true and how would you prove it
The OTM binary is not increasing with volatility. Simply plot the price as a function of implied volatility. eg take
S = 0.99 K = 1 r=0 T =1
and $\sigma$ vary upwards in steps of 0.01 starting at 0. It peaks around 0.04.
Differentiating $d_2$ with respect to $\sigma$ makes this behaviour obvious.