Positive definite Jacobian matrix at any solution to system of equations sufficient for uniqueness?

Question

I have a system of non-linear equations, $F(x) = 0$, with smooth $F : U \to \mathbb{R}^{n}$ where $U \subseteq \mathbb{R}^{n}$ and $U$ is compact, where it's relatively easy to show that 1) a solution exists, and 2) at any solution, the Jacobian of $F$ is positive definite. This thread seems to imply that if the Jacobian of $F$ is positive definite everywhere, then the set of solutions is convex (almost uniqueness). In my case, the Jacobian is not necessarily positive definite everywhere; despite this, are conditions (1) and (2) sufficient for the set of solutions to be convex? This seems true when $n = 1$, but I don't have a good intuition on this problem for $n > 1$.