Continuity conjecture for the Beilinson–Tate residue

Let $K$ be a topological local field over $\boldsymbol{k}$, and let $\operatorname{Res}^{\mathrm{BT}}_{K / \boldsymbol{k}}:\Omega^n_{K / \boldsymbol{k}}\to\boldsymbol{k}$ be its Beilinson–Tate residue functional, where $n=\operatorname{dim}(K)$. The continuity conjecture. The $\boldsymbol{k}$-linear functional $\operatorname{Res}^{\mathrm{BT}}_{K / \boldsymbol{k}}$ is continuous. The continuity of this functional is largely unknown in dimensions $n\geq2$, and is posed alongside the conjecture comparing it with the TLF residue.