Volkswagen AG hasn’t finished a plan for getting 600,000 diesel vehicles off U.S. roads as it faces hundreds of lawsuits for rigging pollution control systems to cheat emissions tests.

U.S. District Judge Charles Breyer in San Francisco, who said at a hearing Thursday that “substantial progress” has been made, set an April 21 deadline for the carmaker to come up with a concrete and detailed proposal. Breyer also remarked that the company faces engineering difficulties in finding a solution.

Volkswagen admitted last year that it had rigged diesel engines on about 11 million cars worldwide so that emission controls switched on only during testing. Those controls shut off while the car was on the road, increasing fuel efficiency and performance but producing nitrogen oxide pollution well in excess of the U.S. legal standard.

The automaker is facing lawsuits by the U.S. Justice Department and state attorneys general and fines of as much as $46 billion, as well as hundreds of private lawsuits, for installing the so-called defeat device.

The case is In Re: Volkswagen “Clean Diesel” Marketing, Sales Practices and Products Liability Litigation, MDL 2672, U.S. District Court, Northern District of California (San Francisco).

–With assistance from Margaret Cronin Fisk and Kartikay Mehrotra.