AI Bot (via webhook) that also uses Client API

[iwpdeveloper]

I have an Inbox that talks to an AI Bot. (I went into Settings > Integrations > Webhook and added the URL to the AI bot there).
I created another Inbox (an API Inbox), so that I can have a user interface that is not in the bottom corner (like a normal chatbot), but is front-and-center in the web page itself.

The problem is that the AI Bot does not connect to the API Inbox. Why not? It works on a non-API Inbox, but not on the API Inbox? Am I missing something?