Integral representation of the invariant hull

Suppose the functional $I$ has the integral form

$$I(u)=\int_\Omega W(\nabla\mathbf{u}(\boldsymbol{x}))\,d\boldsymbol{x}$$

for a continuous integrand

$$W(\mathbf F):\mathbb{R}^{m\times N}\to\mathbb{R}.$$

Let $I_i$ be its invariant hull with respect to an invariant class $\mathcal D_0$, and define

$$W_i(\mathbf F)=\inf_{\mathcal X\in\mathbb{R}^{N\times N}_+} \operatorname{det}\mathcal X\,W\left(\frac{1}{\operatorname{det}\mathcal X}\mathbf F\operatorname{adj}\mathcal X^T\right),$$

where $\mathbb{R}^{N\times N}_+=\{\mathcal X\in\mathbb{R}^{N\times N}:\operatorname{det}\mathcal X>0\}$. Integral representation conjecture. If $I_i$ is itself an integral functional, then $I_i\equiv I_i^*$, namely

$$I_i(u)=\int_\Omega W_i(\nabla\mathbf{u}(\boldsymbol{x}))\,d\boldsymbol{x}.$$

The proposition preceding this statement gives $I_i^*\leq I_i\leq I$, so the assertion identifies the invariant hull with the simpler pointwise invariant density whenever the hull admits an integral representation. The source provides no resolution of this claim.