Artin–Hasse explicit-subspace conjecture

Let $K$ be the local field and let $V$, $J$, $\chi_1$, and $\chi_2$ be as in the paper's local weight recipe. Write $W'(\chi_1,\chi_2)$ for the set of admissible pairs, let $L_V$ be the subspace occurring in that recipe, and let $L_V^{\mathrm{AH}}\subset H^1(G_K,\overline{\mathbf{F}}_p(\chi))$ be the subspace constructed using the Artin–Hasse exponential.

Artin–Hasse subspace conjecture. If $(V,J)\in W'(\chi_1,\chi_2)$ for some $J$, then

$$L_V=L_V^{\mathrm{AH}}.$$

This conjecture gives an explicit description of the subspaces used to determine Serre weights, replacing the $p$-adic Hodge-theoretic description by one based on the Artin–Hasse exponential and local class field theory. The paper states that it would imply the relevant inclusion of local subspaces also when $p=2$; a proof is not supplied.