Let f:{-1,1}^n -> {-1,1} be a monotone , odd ( f(-x)=-f(x) ) Boolean function.

Let F:[0,1]->[0,1] denote the probability that f(x1,...,xn) where x1,...xn are i.i.d. +-1 R.V. with probability p to be 1. 

Our question is the following:

**Is it true that F is convex in [0,1/2] for any such monotone odd f?**

(It is easy to see that F is symmetric around 1/2.)

Another way to view this is that dF/dp is increasing in [0,1/2], meaning the bits become more influential as p is closer to 1/2.

So far, in the example we worked out (such as majority) this turned out to be the case, and we could not find any references, nor managed to prove it or find a counterexample. 
A related fact - when I encountered "graphs" of the probability that certain monotone graph properties are satisfied in G(n,p) as a function of p, they were always "drawn" as convex till the threshold and concave afterwards, but I could not find any explicit references to this phenomena.