Divisibility conjecture for the tame reciprocity map

Let $X$ be a smooth quasi-projective variety over the local field $k$. Write

$$\rho^{\rm t}_X\colon E^{\rm t}(X)\twoheadrightarrow J^{\rm t}(X)$$

for the tame reciprocity map, where $E^{\rm t}(X)$ and $J^{\rm t}(X)$ are the tame class group and tame abelianized fundamental-group target, respectively. Divisibility conjecture. The kernel of

$$\rho^{\rm t}_X\colon E^{\rm t}(X)\twoheadrightarrow J^{\rm t}(X)$$

is divisible. This predicts a refined form of the preceding tame class field theory results; the evidence is discussed in the paper's section on rational curves, but the general assertion is not resolved in the supplied text.