Efficient integration between Twilio and Amazon Connect

We have an AI-based voice agent that needs bidirectional voice streaming and it is served over Twilio programmable voice. One of our customers has a telephony system built using Amazon Connect and they want us to partially handle portions of the calls.

One solution that we explored was to transfer the call, back and forth between Twilio and Connect but that caused two call legs (outgoing and incoming) to remain open on Twilio even when the AI is not engaged anymore, which leads to excessive cost, and also it breaks the call continuity as the call gets completely disconnected from connect when transferred to Twilio.

So we are currently exploring the possibility of handling the transfers over SIP (e.g. SIP invite and refer). However, I have not had any luck in finding comprehensive documentation on how to add SIP endpoint to Amazon Connect. From some Google searches, it seems that it is possible to do it using Chime SDK.

I would appreciate any help or documentation on this. Also, if you have any other suggestions for this problem please let me know!

Thanks;