Injecting proactive message into existing Web Messaging session via Guest API

What’s the correct way to inject a proactive notification into a web messaging session that is already active?

We’re trying to push a message to a customer who has an existing session without triggering a new conversation or requiring the widget to send anything first. The docs for the Guest API are a bit sparse on this specific workflow.

Right now I’m hitting POST /api/v2/conversations/messaging/conversations/{conversationId} with a payload like this:

{
 "from": {
 "id": "system",
 "externalId": "bot-service"
 },
 "to": [
 {
 "id": "guest-session-id-here",
 "externalId": null
 }
 ],
 "text": "Is there anything else I can help with?"
}

The call returns a 200 OK and creates the event, but the message never shows up in the UI. It seems like the guest client isn’t picking up the event. Is there a specific type or direction flag I’m missing? Or do I need to use the WebSocket event stream directly to push this to the client?

Also, if I use the guest endpoint POST /api/v2/messaging/guests/{guestId}/send, it requires the guest to have just started the session. How do we handle mid-conversation proactive nudges?

Any pointers on the SDK method or raw API payload that actually works for existing sessions?