It is possible to start a conversation by sending out a translated outbound message directly.
You will need to create a new email conversation (as shown below, by clicking on Post New Note) on the customer timeline, and give it a subject. Then, set the Activate Unbabel field to true and set the Customer Language.
Be sure to follow the same process as for the regular workflow for outbound messages, using a note and the shortcut. After this, the process remains the same for incoming and outgoing translations.