For clarity and to save space remove the timezone in brackets e.g. (EDT) from the user card. Also add a title to the user time span to say it is Local Time.
The local time was not updating between user cards because the computed property was not used correctly.
There's an old saying in Tennessee — I know it's in Texas, probably in Tennessee — that says, fool me once computed properties, shame on — shame on you. Fool me — you can't get fooled again.
This adds a site setting (default off) to optionally show a user's local time and timezone in their user card. For example, I live in Brisbane, and if at 3:30PM my time I were to open a user who lives in California's card I would see 22:30 (PST).