Web Messaging SDK: Send proactive message to existing session ID

Trying to push a proactive notification to a customer who already has an active web messaging session. We’ve got the session.id from our CRM integration. The docs show how to start a new conversation, but nothing on injecting a message into an existing one without the user typing first. Tried hitting POST /api/v2/webmessaging/conversations/{conversationId}/messages with the guest token, but it’s throwing a 403 Forbidden. Seems like the guest token doesn’t have permission to write to the conversation directly. Is there a specific admin endpoint or a different SDK method to trigger this? The payload looks standard enough:

{
 "text": "Your support ticket has been updated.",
 "type": "text"
}

Stuck on the auth part.