Genesys Web Messaging: Triggering proactive notification to existing guest session

Trying to send a proactive notification to a guest who had a web messaging session 20 minutes ago. The session is still active in the platform, but I can’t find a direct API endpoint to push a message to that specific conversationId from our external CRM.

I tried posting to /api/v2/conversations/messaging/ with the existing ID, but I get a 403 Forbidden. The docs for the Guest API are sparse on this. Is there a specific header or a different endpoint for injecting messages into an open session via server-to-server call?