Extremal field implies minimality --- low regularity assumptions?

Question

In A stability theorem for minimal foliations on a torus, Moser is studying variational integrals of the following form: \begin{equation*} \mathcal{F}(u) = \int F(x,u(x),Du(x)) \, dx \end{equation*} Here the argument $u$ is a scalar function. It shouldn't be particularly relevant to my question, but $F(x,u,p)$ is $\mathbb{Z}^{d + 1}$-periodic in $(x,u)$ for each $p$. Also, $F$ is smooth and uniformly elliptic.

Specifically, Moser is interested in "minimal foliations" corresponding to $\mathcal{F}$ defined like so:

A function $u \in C^{r}(\mathbb{R}^{d} \times \mathbb{R})$ is a $\mathbb{Z}^{d + 1}$-invariant $C^{r}$-minimal foliation if

1. For each $\lambda \in \mathbb{R}$, the function $u(\cdot,\lambda) : x \mapsto u(x,\lambda)$ is an extremal of $\mathcal{F}$ (i.e. a solution of the Euler-Lagrange equation)
2. For each $x \in \mathbb{R}^{d}$, the function $\lambda \mapsto u(x,\lambda)$ is a $C^{r}$-homeomorphism of $\mathbb{R}$ onto itself
3. The foliation given by the leaves $x_{n + 1} = u(x,\lambda)$, $\lambda \in \mathbb{R}$, is invariant under the $\mathbb{Z}^{d + 1}$ action

(The $\mathbb{Z}^{d + 1}$-action is given by $T_{\bar{k}}u = u(\cdot - k) + k_{d + 1}$ if $\bar{k} = (k,k_{n + 1}) \in \mathbb{Z}^{d + 1}$.)

Now Moser claims that the family of extremals $\{u(\cdot,\lambda)\}_{\lambda \in \mathbb{R}}$ are not only extremals of $\mathcal{F}$, but actually Class A minimizers. He writes (pp. 253):

"The extremals representing the leaves of such a minimal foliation are special solutions of the Euler equations. They minimize the functional $\mathcal{F}$ taken over any large ball, compared to any other admissible functions with the same boundary values. This follows from the fact that leaves of such a foliation can be viewed as a 'field of extremals' in the sense of calculus of variations. It is well known that every extremal which can be embedded into such a field of extremals is minimal provided... [$\mathcal{F}$ satisfies an ellipticity condition]. In other words, a field of extremals is always a field of minimals."

When $r \geq 3$, it seems to me Moser's claim can be justified by the results in Chapter 6, Section 3 of the two volumes by Giaquinta and Hildebrandt.

Is this really well-known when $r = 0$? If so, can someone provide a reference? (Note that the maps $u(\cdot,\lambda) : x \mapsto u(x,\lambda)$ are automatically smooth, being extremals, so the regularity exponent $r$ is only really relevant to the variable $\lambda$.)