= Solution
Let $u,v$ be two <critical points> with the same classical boundary values, and put $w=u-v$. Since $F$ is independent of $z$, the <Euler-Lagrange equation> for each is $D_iF_{p_i}(x,Du)=0$. By the <fundamental theorem of calculus> along the segment between the <gradients>,
$$
F_{p_i}(x,Du)-F_{p_i}(x,Dv)=A_{ij}(x)D_jw,\qquad A_{ij}(x)=\int_0^1F_{p_ip_j}(x,Dv+tDw)\,dt.
$$
Subtracting the equations gives $D_i(A_{ij}D_jw)=0$. The <uniformly convex variational integrand> makes $A$ symmetric and <uniformly elliptic> with the same positive lower bound. Since $F$ is smooth and $u,v\in C^2(\overline\Omega)$, $A$ is continuously differentiable locally in $\Omega$, so
$$
A_{ij}D_{ij}w+(D_iA_{ij})D_jw=0
$$
is a classical <nondivergence-form elliptic operator> with no zeroth-order term.
To avoid assuming that derivatives of $F$ stay bounded all the way to the boundary, apply the proved <weak maximum principle> on inner open sets $\Omega_\delta=\{x\in\Omega:\operatorname{dist}(x,\partial\Omega)>\delta\}$. Their closures lie compactly in $\Omega$, so all linearized coefficients there are bounded. The maximum principle applies on each component and yields
$$
\sup_{\Omega_\delta}|w|\leq\sup_{\partial\Omega_\delta}|w|.
$$
Every point of the inner boundary is at distance $\delta$ from $\partial\Omega$. Since $w$ is continuous on the compact closure and equals zero on that boundary, the right-hand side tends uniformly to zero as $\delta\downarrow0$. Every fixed interior point eventually belongs to $\Omega_\delta$, proving
$$
\boxed{u=v\text{ in }\Omega.}
$$
This establishes <uniqueness for uniformly convex gradient Dirichlet problems> using precisely the boundary continuity and ellipticity needed for the comparison.
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