Branding - Custom Admin Bar

We have enabled custom admin bar within the branding plugin settings, to provide our users with custom navigation links throughout the entire network of sites.
Some of these sites are public within the network (i.e. with privacy setting 'Visitors must have a login - anyone that is a registered user of the network can gain access.') so users have no specific rol on these sites.
However, when our users go to these sites within the network, our custom admin bar dissappears because they have no role, making it more difficult for them to navigate around the network of sites.

We would like to be able to display the custom admin bar for our users on all sites within the network, even on sites where users have no specific role assigned.

Thanks in advance!