PATCH /api/v2/conversations/calls/{id} returns 409 Conflict on transfer

Trying to move a call to a different queue using the Conversations API. The request fails with a 409 Conflict error. Here is the JSON body I’m sending:

{
 "to": {
 "id": "new-queue-id",
 "type": "queue"
 }
}

The interaction is in ‘connected’ state. Is there a specific permission I’m missing?