Gauge-rigidity conjecture for the Calderón problem for connections

Let $(M,g)$ be a Riemannian manifold, let $E$ be a Hermitian vector bundle over $M$, and let $A$ and $B$ be unitary connections on $E$. Denote by $\Lambda_A$ and $\Lambda_B$ the Dirichlet-to-Neumann maps associated with the connection Laplacians determined by $A$ and $B$. A gauge equivalence is a fibrewise Hermitian-preserving bundle automorphism of $E$; it is boundary-trivial when its restriction to $\partial M$ is the identity. Gauge-rigidity conjecture. The equality

$$\Lambda_A=\Lambda_B$$

holds if and only if there exists a boundary-trivial gauge equivalence that pulls back $B$ to $A$. This asserts that boundary-trivial gauge transformations are the only obstruction to recovering a unitary connection from its Dirichlet-to-Neumann map. The general Calderón problem for connections remains open, although partial results are known under additional geometric and analytic assumptions.