The British press is arguably under-regulated and over-legislated. The Press Complaints Commission was never a regulator in any meaningful sense of the term. The rules on ensuring plurality of voice don't work. Meanwhile, the country has a set of laws that can chill the very thing they should encourage – let's call it public-interest journalism – without any strong countervailing assertion of press freedom such as is enshrined in the first amendment of the US constitution.

Lord Justice Leveson begins the final leg of his inquiry into the press by hearing from those who have proposals for improving the situation, including the industry itself, represented by Lord Black, and from the man charged with reforming the PCC, Lord Hunt. He will also hear from academics and pressure groups. Lords Black and Hunt have done much to secure a large measure of industry consensus behind a new form of more independent regulation with greater powers of investigation and punishment. The system would rely on the major publishers entering into long-term contracts with the new regulator, and would be non-statutory. This is real progress.

But if regulation is to be voluntary, how do you prevent publishers simply walking away, as Richard Desmond, the owner of the Express titles, did with the PCC? One idea – which looks slightly protectionist and may not be workable in the blogging age – is to limit information, press conferences and events to accredited staff of large publishers. Another – more desirable, anyway – is to create an arbitral wing of the regulator, which would deal with libel and privacy matters swiftly and cheaply. The courts could be obliged to take into account attempts to resolve disputes via adjudication or mediation. Costs and damages could be lower for those who were part of the system.

Between now and finishing his report in the autumn, Lord Justice Leveson will have to use the inquiry's summer break to sort out many things. Are these carrots sufficiently convincing? Does he find any of it ambitious enough? Does any of it deal with his instruction to think about plurality? If he's unpersuaded by the incentives to sign up, will he argue for some form of statutory backstop? Are there greater wins for the press – on libel, on a British first amendment – that would also require some form of statute? How, Leveson will be wondering, would his proposals knit with the director of public prosecutions' wholly positive thinking on harmonising public-interest defences against criminal charges? Will the Leveson recommendations be future-proofed in relation to digital publishing? Finally, what are the politics of all this? Michael Gove, who has made much of his friendship with, and admiration for, Rupert Murdoch, has staked out a libertarian view of minimalist regulation. There is a great gulf between that position and others in parliament who consider that the measures currently proposed by the press don't go far enough.

Lord Justice Leveson has cautioned against reading too much into his interventions as the inquiry has progressed. But there have been a few radical signals. He is evidently pondering doing something about our onerous, expensive and embarrassing libel laws. That would need a bit of statute – if only an amendment to the current defamation bill. He has talked about a balancing bit of law – an amendment to enshrine the independence of the press as section 3(1) of the Constitutional Reform Act enshrines it for the judiciary.

Most of the press would need persuading that any form of statute is desirable or necessary. Leveson, however, may want to show that he can not only regulate more effectively, but also propose a new settlement protecting public-interest journalism. That – combined with the DPP's proposals – could mark something of a watershed for British journalism, particularly if the judge says something strong on plurality. The Leveson inquiry has revealed shocking things about press, policing and politics. It retains the potential to achieve something far-reaching and important.