Waymo supposedly processes their lidar signals to detect other cars with reasonable accuracy even in snow/rain(https://youtu.be/UrJ4-AUL4U0?t=9m41s). They only operate in pre-mapped areas so they should be able to follow lanes just fine even if the actual markings are obscured (by weather or by poor maintenance).
This kind of approach (mapping everything beforehand and using lidar for perception) is used by other players as well, and seems the most likely to work for short term level 4. Trying to do everything without the benefits of lidar and high-res maps (what e.g. Tesla are doing) is a more general solution, and cheaper hardware-wise, but will take longer to get right.