We would need to have a distinct difference between "STOPPED" and "IDLE"

imho the diff between stopped and idle is the same from the users perspective if i switch into the convo, it loads instantly and i resume where ever i left it would nice to not introduce too many different states as to simplify the experience maybe just active/non-active would suffice, and the convo pane can filter a view based on this

Yes

The hide conversations older than 1 hour is a great option if you have not used it. (Ignore the first option being highlighted in my screenshot)

That does satisfy this mostly - I do think we need a way to search conversations in the long run though