Let $F$ be a $n$-dimensional local field. If $n=0$ or $1$, the topological structure on $F$ was well-known, however if $n>1$ i.e, $F$ is a higher dimensional local field, I don't know something nice topological structure on $F$. Matthew Morrow introduced so-called "higher topology" on the higher dimensional local fields in his survey https://arxiv.org/abs/1204.0586, but this "higher topology" does not provide the structure as topological field with $F$. Indeed he described that any fixed element $\alpha\in F$, multiplication $$\alpha \times\colon F\longrightarrow F ~;~ \beta \longmapsto \alpha\beta$$ are all continuous map in higher topology. Unfortunately, this property is weaker than the definition of topological ring. So we want some topological structures as that addition and multiplication $$ +,\times\colon F\times F\longrightarrow F $$ are continuous, and compatible with that of residue fields. I.e, In these topologies, for the ring of integer $\mathscr{O}_{F}$ of $F$ with relative topology, the canonical surjection $\mathscr{O}_{F}\longrightarrow F_{n-1}$ should be continuous and open morphism, where $F_{n-1}$ is the residue field of $F$ equipped with this topology.

I think that the attempt to give $F$ to such a topology have so for been unsuccessful up on now. I know Fesenko, Parshin and Camara are challenging this experiments. but these does not also seem to work.

Question. Are there exist the topological structures on higher dimensional local fields satisfying some properties as above? or do you know about some related research?