I would say 'no' officially but they should still find ways of commemorating the history as belonging to the city, even if it doesn't belong to the current franchise.
To take it to the more 'concrete' level, keep the Stanley Cup banners but don't have the modern logo on them. Would also love to see them sporting the barberpoles as a turn-back-the-clock or third jersey (there's been many threads on that). Not having the history officially part of the current team doesn't take away from the significance of the history and the importance of keeping it alive.
I don't think the current team leverages the history enough (instead we get Centurions with dodgy microphones and helmets that don't fit
).