U.S. District Judge Charles Breyer said that if no highly specific proposal is presented to him at that time—explaining precisely which vehicles could be repaired by what dates and which others, incapable of being adequately repaired, would have to be bought back from consumers by Volkswagen—then he would schedule a trial that he would adjudicate (sitting without a jury) for this summer.

He granted the parties a one-month extension—he had previously ordered them to have a concrete proposal ready by Thursday—in light of the “substantial progress” they had made, according to what he had learned from the settlement master he appointed earlier in the case, former F.B.I. Director Robert Mueller.

Judge Breyer—who is the brother of U.S. Supreme Court Justice Stephen Breyer, and sounds exactly like him—is presiding over more than 500 consolidated class action suits from all over the country that were filed after the Volkswagen diesel scandal broke last September. The suits raise allegations of consumer fraud, unfair competition, and false advertising.

Also before Breyer is a suit brought jointly by the U.S. Environmental Protection Agency and Department of Justice for environmental violations, where maximum fines could theoretically reach $45 billion (though no one expects a fine so high). All 50 state attorneys generals are also pursuing cases.

On September 3, after 15 months of less than candid discussions with regulators at the California Air Resources Board, Volkswagen admitted that it had installed so-called cheat software on hundreds of thousands of its diesel cars in the U.S. Such software—known as a defeat device—tricks regulators by instructing the vehicle’s exhaust equipment to behave in a way that will satisfy emissions standards when the car is undergoing testing in a controlled lab setting, though the same vehicle produces illegal levels of pollution when the car is operating in real-world driving conditions.

According to the E.P.A. and the California Air Resources Board (CARB)—a key auto emissions regulator—the VW diesels in question emitted up to 40 times the permissible levels of oxides of nitrogen, or NOX, when operating outside the test lab.

The cars involved in the scandal (which include some produced by Volkswagen’s Audi and Porsche units as well) encompass model years 2009 to the present and comprise four main diesel engine categories: three generations of 2.0 liter engines as well as tens of thousands of 3.0 liter diesels as well. The technical challenge of bringing a particular car into compliance differs depending on the engine category it falls into.

The viability of any repair is affected not just by the cost to Volkswagen but also by the adverse impact the repair may have on fuel economy and driving performance. All repairs must be approved by the EPA and CARB.

Judge Breyer, who grew up in California and has alluded in court to seeing the noxious smog blanketing Los Angeles as a child, has repeatedly stressed the need to move quickly. “It’s an ongoing harm that must be addressed,” he said at a hearing in February. “They are polluting and, therefore, we must address it.”

NOX emissions contribute to the production of smog, a condition that aggravates, and may even cause, respiratory ailments like asthma and emphysema. Regulators say they also contribute to the production of fine particulate matter associated with premature death.

In a statement issued after the hearing, Volkswagen said: “Volkswagen is committed to resolving the U.S. regulatory investigation into the diesel emissions matter as quickly as possible and to implementing a solution for affected vehicles, as we work to earn back the trust of our customers and dealers and the public. We continue to make progress and are cooperating fully with the efforts undertaken by Judge Breyer, working through Director Mueller, to bring about a prompt and fair resolution of the U.S. civil litigation.”

Volkswagen’s lead counsel at the hearing was Robert Giuffra of Sullivan & Cromwell, who told Judge Breyer that he had been working “round the clock” on the remediation plan for the vehicles, to the point where last month “had been the hardest month of my now 28-year career.” Breyer gave no hint of letting up on him, however.

The lead plaintiffs lawyer, heading a 22-attorney plaintiffs steering committee, is Elizabeth Cabraser of San Francisco’s Lieff Cabraser Heimann & Bernstein.

Judge Breyer is able to threaten to hold a trial without a jury because many of the California plaintiffs are suing under the state’s Unfair Competition Law. That statute does not carry a right to trial by jury because it does not authorize the award of true “damages.” It does, however, permit a judge to issue orders, and those orders could include requiring Volkswagen to return money to consumers (restitution) that they have previously paid.

As a result of the scandal, Volkswagen has postponed the announcement of its 2015 annual results until April 28, and has put off its shareholder meeting until June 22.

The company has also said it would announce the preliminary results of an internal investigation it commissioned by the American law firm of Jones Day in mid-April.

My colleague Geoff Smith and I published a feature about the toxic culture that led to the Volkswagen clean diesel debacle, called “Hoaxwagen,” last month.