The tame-base-extension limit conjecture for Borger's Artin conductor

Let $E$ be the local field under consideration and let $G_E$ denote its absolute Galois group. Let $\rho$ be a representation of $G_E$ with finite local monodromy. For a positive integer $m$, let $E_m/E$ be an extension that is tamely ramified of tame degree $m$, and write $G_{E_m}$ for its absolute Galois group. Let $b'(E_m)$ be Borger's Artin conductor of the restriction of $\rho$ to $G_{E_m}$. The tame-base-extension limit conjecture. Then

$$\limsup_{m\to\infty}m^{-1}b'(E_m)$$

equals the differential Swan conductor of $\rho$.

This conjecture proposes a reconciliation between Borger's Artin conductor and the differential Swan conductor in the imperfect-residue-field setting: forcing good behavior under tame base extension should recover the differential invariant. Its resolution is not given here.