The elliptic-cone higher-integrability conjecture for PDE-constrained measures

Let $c9$ be an open set, let $V$ be a finite-dimensional vector space, and let bc otin\text{\mathcal{M}}(c9;V) satisfy the PDE constraint

$$\mathcal{A}\mu=0.$$

Write $\frac{\mathrm{d}\mu}{\mathrm{d}|\mu|}(x)$ for its polar, let $\Lambda_{\mathcal{A}}$ be the wave cone of the differential operator $\mathcal{A}$, and let $\mathcal{K}\subset V$ be a convex cone. The cone is elliptic when

$$\mathcal{K}\cap\Lambda_{\mathcal{A}}=\{0_V\}.$$

The elliptic-cone higher-integrability conjecture. If

$$\frac{\mathrm{d}\mu}{\mathrm{d}|\mu|}(x)\in\mathcal{K}$$

for $|\mu|$-almost every $x\in\Omega$, then analogous $\mathrm{L}^p$ estimates to those in Theorem should hold for $\mu$.

This conjecture was explicitly identified in the source as false; the paper explains that its perturbative results apply only to a small conical neighborhood of a linear space, rather than to every closed convex cone intersecting the wave cone only at the origin.