You do not have the Browse users and groups global permission is correct. The user picker that shows suggested users depends on the global permission to browse users and groups and without that permission the suggestion list will remain empty.

The Browse users and groups global permission is a site level permission that controls whether an account can search for and see other users and groups. If you do not have it then the picker cannot return matches even if you can otherwise manage the board. Granting that global permission or asking someone who can manage global permissions to grant it will enable the suggestions.

You lack a project role on the project associated with the board is incorrect because project role membership does not affect the global user search that powers the suggestion list. The picker relies on global visibility of users rather than on project role membership.

Refreshing the browser will resolve the issue is incorrect because a browser refresh will not change your account permissions. The absence of suggestions is caused by a permission setting and not by a temporary UI glitch in most cases.

The board currently has no administrators assigned is incorrect because the presence or absence of other board administrators does not prevent the user picker from suggesting users. The picker behavior is controlled by your permission to browse users and groups.

You are not a site administrator is incorrect because being a site administrator is not required just to see user suggestions. You only need the specific Browse users and groups global permission which can be granted without making someone a site administrator.

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