The compatibility conjecture for Beilinson–Tate and TLF residues

Let $K$ be an $n$-dimensional topological local field over $\boldsymbol{k}$. Write $obreak\Omega^n_{K / \boldsymbol{k}}\twoheadrightarrow\Omega^{n,\mathrm{sep}}_{K / \boldsymbol{k}}$ for the canonical continuous surjection, and let $\operatorname{Res}^{\mathrm{BT}}_{K / \boldsymbol{k}}$ and $\operatorname{Res}^{\mathrm{TLF}}_{K / \boldsymbol{k}}$ be the Beilinson–Tate and TLF residue functionals, respectively. The compatibility conjecture. The diagram of $\boldsymbol{k}$-linear homomorphisms commutes, equivalently

$$\operatorname{Res}^{\mathrm{BT}}_{K / \boldsymbol{k}}=\operatorname{Res}^{\mathrm{TLF}}_{K / \boldsymbol{k}}\circ\bigl(\Omega^n_{K / \boldsymbol{k}}\twoheadrightarrow\Omega^{n,\mathrm{sep}}_{K / \boldsymbol{k}}\bigr).$$

The conjecture is known for dimensions $n\leq 1$: it is trivial for $n=0$ and follows from Tate's original work for $n=1$. The higher-dimensional case motivates the review of the TLF residue and its properties in the paper.