Quasi concavity of utility function
Your conjecture seems to be contradicted, at least for small values of $\sigma$. 
You can draw the function with the following R-code:  

qq_f = function(x,k,h,sig){   
-pnorm(-k, sd=sig)*( (dnorm(h*(1-x), sd=sig))^2 ) - 0.5*dnorm(-k^2, sd=sig)*( 2*pnorm(h*(1-x), sd=sig) -1 )^2  
}  
curve(qq_f(x,k=0,h=1,sig=0.5),col='blue',xlim=c(-1,3),type='l',main="A candidate quasi-concave function")