Resolved
As of approx 09:00 UTC today, restarting Safari 26 on MacOS / iOS fixes the issue of the messenger not working in incognito mode or when "Advanced Tracking and Fingerprinting Protection" is enabled. We are following-up with Apple.
Identified
Safari 26 has broken the Intercom messenger when "Advanced Tracking and Fingerprinting Protection" is enabled, either by using Safari's private browsing mode or explicitly enabling it in Safari's settings. The Intercom messenger is usable on Safari 26 by default. We are working on working around the issue, and are communicating with Apple.