Show that F can have at most two fixed points

Question

Consider a function $F:\mathbb{R}^n\to\mathbb{R}^n$, where $F=(F_1,...,F_n)$. Suppose that $F$ is strictly quasi-concave, and that for all $i=1,...,n$ the function $F_i:\mathbb{R}^n\to\mathbb{R}$ is increasing, in the sense that $x_i\geq y_i$ for all $i$, implies $F_i(x)\geq F_i(y)$. Finally, assume that $F(0)>0$ (i.e. $F_i(0)>0$ for all $i$). Is it true that $F$ can have at most two fixed points? One positive, and another one negative?