How to clear your "My Closed Conversations"
If you are cleaning up your inboxes and noticed that you can't "clear" any closed conversations under the "My Closed Conversations" tab, it is due to the fact that you are technically still a participant in the conversation. In order to clear these conversations you will have to leave the conversation in this case.
So for example a trick for this is that you can essentially re open the conversation and then click the "close+leave" command. This will automatically remove you from the conversation and will also close it out! Take a look here for an example.
Another alternative to this could be to access the participants tab on the right hand side in your conversation details and click the 'x' next to your name to leave the conversation as well!