Let $f_i: \mathbb{R}^{Nn} \rightarrow \mathbb{R}$ be convex and twice continously differentiable functions for every $i=1...N$. Consider the function $F: \mathbb{R}^{Nn} \times \mathbb{R}^{Nn}$ defined as follows $$F(x_1,...,x_N)=\begin{bmatrix}\nabla_{x_1}f_1(x_1,...,x_N)\\\nabla_{x_2}f_2(x_1,...,x_N)\\...\\\nabla_{x_N}f_N(x_1,...,x_N)\end{bmatrix}\,,$$
where $x_i \in \mathbb{R}^n$ for every $i=1...N$. Is the Jacobian of $F$, $JF \in \mathbb{R}^{Nn \times Nn}$, always positive semi-definite for any choice of functions $f_i$? If not, could you provide a counter-example?