Conjecture on tame local monodromy at zero

Let $\rho_k$ be the local-system representation, let $I_0^{\operatorname{tame}}$ be the tame inertia group at $0$, and let $\kappa\in\widehat{T}[2]$ be the element defined in the source's discussion of $\kappa$. A topological generator of $I_0^{\operatorname{tame}}$ is mapped to an element conjugate to $\kappa$.

Local monodromy conjecture at zero. Under $\rho_k$, a topological generator of the tame inertia group $I_0^{\operatorname{tame}}$ gets mapped to an element conjugate to $\kappa\in\widehat{T}[2]$. Equivalently, the unipotent part of the local monodromy at $0$ is trivial.

Together with the conjecture on monodromy at infinity, this is stated to imply the claimed identification of the local system in type $G_2$ with the one constructed by Dettweiler and Reiter. The supplied text does not state whether the conjecture has been proved or refuted.