After chatting with Ferry (who was wonderful!), we determined there is currently no functionality to rearrange/pin default attributes listed in the conversation attributes. It would be nice if we could do this with any of the ones that are listed.
In my particular case, we were trying to get the “Language” attribute, which currently shows under “See all [conversation attributes]” to show under the “Sentiment” attribute for easier visibility. Ferry pointed out that the order of some default fields in the details panel aren’t configurable. We’re able to choose whether they’re visible, but their position is fixed. Language is one of those default fields impacted.