Non-polyconvex $Q$-integrands with lower semicontinuous energies

For any fixed $Q \in \mathbb N$, we construct a positive measure on the space of positively oriented $2$-vectors in $\mathbb R^4$, whose barycenter is a simple $2$-vector, yet which cannot be approximated by weighted Gaussian images of Lipschitz $Q$-graphs. The construction extends to positively ori