Make sure your LinkedIn contact is synced to HubSpot with Hubris.
If you haven’t done this already, click on the Hubris button on their profile, then click ‘Sync with HubSpot’.
This will ensure that information from their LinkedIn profile is synced to their HubSpot contact record (if they don’t have a contact record, HubSpot will create one).
Open your conversation with your LinkedIn contact.
Please note that Hubris LinkedIn message sync will only work for the 20 most recent contacts you’ve messaged, so you may not be able to save a conversation that’s been dormant for a while.
Click ‘Sync with HubSpot’ to save the conversation against the contact’s activity timeline.