I created a publicly shared Web Experience using ArcGIS Online Experience Builder. The web map is also publicly shared. I also enabled Anonymous Access to my AGOL organization. The experience map basmap tool shows organization basemap gallery to signed in users, but shows a different set of basemaps for anonymous users. The problem does not occur for web applications built by Web AppBuilder.
Here are the screenshots:
Basemap tool shows Organization defined basemaps for signed in users:

Basemap tool does not show organization basemaps for anonymous users:

Does anyone else have this problem? Any suggestions would be appreciated.