Kummer pro-étale extension for logarithmic topological constructions

Let $X$ be a smooth space in the setting of the paper. The Kummer pro-étale site $X_{\text{Kummer-pro-étale}}$ has logarithmic perfectoid subdomains as a basis of neighbourhoods.

Kummer pro-étale extension conjecture. The construction developed in the section should extend to suitable Kummer pro-étale sites with logarithmic perfectoid subdomains forming a basis of neighbourhoods for the topology.

This conjecture is motivated by the cited foundations for logarithmic and pro-étale sites. It is a foundational extension claim, and the supplied text gives no proof or resolution.