I’d try to implement that as an “inventory actor.” I’d make this actor be “always relevant” in network properties.
The “inventory actor” would have the list of items (of course!) as a replicated property, as well as a “is showing widget” replicated property, and “currently controlling player” replicated property.
When someone requests the inventory, they send an event to that effect to the server, and the server does the “who gets to control it” determination, and then sets the “inventory is open” flag.
I’d use RepNotify to make the actor show the appropriate widget when that flag is set on each client.