Gauge uniqueness conjecture for the nonabelian twisted X-ray transform

Let $(M,g,bb)$, with $bb\in C^\infty(SM)$, be a compact Riemannian surface with strictly $bb$-convex boundary and nontrapping $bb$-geodesic flow. Let $\mathcal{G}$ be the gauge group of smooth maps $u:M\to G$ satisfying $u|_{\partial M}=\operatorname{Id}$, acting on pairs by $(A,\Phi)\cdot u=(u^{-1}du+u^{-1}Au,u^{-1}\Phi u)$. For $A,B\in\Omega^1(M,\mathfrak{gl}(n,\mathbb{C}))$ and $\Phi,\Psi\in C^\infty(M,\mathfrak{gl}(n,\mathbb{C}))$, let $C_{A,\Phi}^{\lambda}$ denote the nonabelian twisted X-ray transform.

Gauge uniqueness conjecture. If

$$C_{A,\Phi}^{\lambda}=C_{B,\Psi}^{\lambda},$$

then there exists $u\in\mathcal{G}$ such that $(A,\Phi)\cdot u=(B,\Psi)$.

This conjecture asserts injectivity of the nonabelian twisted X-ray transform up to the natural gauge. It was settled for $\lambda=0$ under the absence of conjugate points, while the general $\lambda$-geodesic-flow case remains open.