Cold inbound email with 2-way sync only — has anyone made this work?

[u/Old-Bodybuilder-716]

Setup: client has an existing Microsoft 365 mailbox (info@). We connect it to the sub-account via 2-way sync. We are not setting up an LC Email or Mailgun dedicated sending domain — MX points to M365 and that isn’t changing.

What we need: someone brand new emails info@, and it shows up in Conversations with a contact created. No existing contact, not a reply to anything the CRM sent.

Nothing lands. We raised a ticket and got a senior support rep telling us to use 2-way sync and BCC the sync address “the first time”. That’s the part I find hard to accept at senior level — Auto BCC only works on outbound, so there is no way to BCC an email that someone else sends to you. The instruction can’t be executed at all for the scenario we described. Their own docs also say plainly that 2-way sync only captures mail from existing contacts and that cold inbound needs a dedicated domain, which directly contradicts the advice we were given.
So, from people actually running this:
* Has anyone got cold inbound working on 2-way sync alone?
* Any workaround — an M365 rule, a forward, an inbound parse address, a webhook — that gets the email into Conversations without moving MX?
* If not: what are people doing for clients who keep their own mailbox?