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

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 for any fixed $Q\in \mathbb{N}$. The construction extends to positively oriented $m$-vectors in ${\mathbb{R}}^n$ whenever $n-2\ge m\ge 2$. This geometric obstruction implies that the approximation result established in [Arch. Ration. Mech. Anal., 2025] is sharp: all $Q\in \mathbb{N}$ are indeed necessary to ensure the density of weighted Gaussian images of Lipschitz multigraphs in the space of positive measures with simple barycenter. As an application, we prove that for every $Q\ge 1$ and $p\ge 2$ there exists a non-polyconvex $Q$-integrand whose associated energy is weakly lower semicontinuous in $W^{1,p}$. This also provides new insight into the question posed in [Arch. Ration. Mech. Anal., 2025, Remark 1.14].