The sharp Sobolev injectivity conjecture for Schrödinger potentials

Let $d/2\leq p<\infty$, let $\Omega$ be a bounded domain with Lipschitz boundary, and let $H_c^{s,p}(\Omega)$ denote the compactly supported Sobolev space of potentials. For a potential $q$, write $\mathcal L_q$ for the associated boundary measurement map. Injectivity conjecture. If $q_1,q_2\in H_c^{s,p}(\Omega)$ satisfy

$$\mathcal L_{q_1}=\mathcal L_{q_2},$$

and

$$s\geq s_d^*(p):=\max\left\{-1,-2+\frac{d}{p}\right\},$$

then $q_1=q_2$. This interpolates the known injectivity result for $q\in L^{d/2}$ with the conjectured higher-dimensional extension of Haberman's $H^{-1,d}$ result; the source presents it as a natural conjecture for the Calderón problem.