The Tasks panel on the 360 Dashboard View
The Local time column in the Tasks section on the 360 Dashboard View will always reference and calculate from the Prospect's Time Zone field (not to be confused with their Local Time tile).
If you click on the Contact from that task view list and view the prospect record, once you click Edit you will find the Time Zone field under the Location section. If a time zone exists, you can see the time zone being populated under the prospect's profile page under the contact section on the right side.
If you're curious on the difference between the Prospect's Time Zone field and Local Time tile, as well as how those fields are populated, please see our support article here.