Hey there,
I am really sorry for the delay of this reply as it is a weekend for us here.
Let me try explain the current default behaviour regarding this :
If that is normal registered user, when they pinned their stream item, it will show this pinned stream item on their profile page, mean this pinned item will always show at the top on their profile page.
If that is Superadmin pinned those stream item on the site, it will always show at the top on dashboard page and everyone who registered on your site, they can able to see this pinned stream item on the dashboard page.
Most of the cases, if that is Superadmin pinned stream item on the site, mean other Superadmin should be agree with him to do that, if this Superadmin always pinned those weird stream item on the site for personal purpose, i think you should take action and give him some warning regarding this.
Imagine if we allow all the site Superadmin to have this permission to unpin other Superadmin pinned stream item, it will become out of control.
Regarding with your case, this 'Robert Sullivan' is one of your site Superadmin, if this user doesn't moderate on the site, i would suggest you remove this user superadmin privileges.
If you still want all the superadmin to have this permission, then you have to modify the core file in order to achieve this.
JoomlaFolder/administrator/components/com_easysocial/includes/stream/item.php
You can refer on my attached screenshot which line of code you have to inject this following code.
if ($this->my->isSiteAdmin()) {
return true;
}