You can monitor the conditions on the passes here:
Most in The Heartland remain closed for now. The Gotthard (about 27 May) is a priority, and they have about 200,000 TONS of snow to move there. Next come the Oberalp (about 14 May) and the Klausen (about 5 June). Then they will work on the Furka (5 June), Susten (19 June), and Grimsel.