I've logged a case with Genesys Support over this but I thought I'd reach out here too.
This is a bit of an edge case but if a user logs into Genesys Cloud using a browser e.g. Chrome, and that Organisation has domains set as part of the Restrict Genesys Cloud Embedding organisation settings, if the user then at some point in the future, unrelated to their use of Genesys Cloud, uses the same browser to navigate to a web site that uses Genesys Web Messaging, the messenger will not display unless that other web site's domain is listed in the Genesys Cloud organisation settings. This could be any web messaging deployment which is part of any Genesys Cloud install - not the one they originally logged into.
This appears to be because the use of Restrict Genesys Cloud Embedding creates/extends a cookie with values such as: GENESYS-Domain-List and __Secure-GENESYS-Domain-List.
The web messaging code seems to reference these cookies and will not render on the other web site. It is blocked due to CSP (Content Security Policy).
Given that messenger deployments have their own mechanism to restrict domains, I don't see why these cookies should be referenced by web messaging as well.