Quasiconcavity for the sum of specific quasiconcave functions

I want to show that a function , which is separable (additively decomposed) in two quasiconcave functions, is also quasiconcave (QC). I know that the sum of QC functions is not generally QC, but my numerical simulations seem to suggest that is.

I have tried using the definition (i.e. trying to show that it has convex upper contour sets by plugging ... etc), but I can not prove the inequality. I also tried using the border Hessian, but i cannot sign the determinant.

Any other ideas??

Below is a more detailed description of the problem:

Where and , and

for

and the functions F(.), Bl(.), Bh(.) and P(.) are strictly increasing. Additionally, Bl''>0, and Bh''<0.

F'' and P'' can be positive or negative (as required to guarantee quasiconcavity --my simulations suggest that only P''>0 is necessary).

I would appreciate any hints --including other forums where I could post this question