Scientists pored over a series of pressure tests from BP's well in the Gulf of Mexico today, trying to discover whether engineers had unknowingly already sealed off the gusher for good.

A family strolls along an oil-soaked public beach in Gulf Shores, Alabama. Photograph: Dan Anderson/EPA The tests could decide whether the official epitaph for the Macondo is written tomorrow or sometime next week, when crews were scheduled to complete two relief wells that officials have described as the last step in permanently securing the BP well.

A decision to stop work on the relief wells would bring an unexpected conclusion to the Deepwater Horizon crisis in the Gulf, which has devastated the local economy and environment, cost BP billions, and shaken confidence in Barack Obama's leadership.

It might also be a hard sell to a public which has been told repeatedly that the relief wells are the only sure way of plugging the well for good.

In Venice, a fishing port which saw some of the worst oil slicks, the prospect of Obama administration officials and BP executives declaring an early end to the well was greeted with suspicion. "There's plenty of oil still out there," said one of the airboat operators who have been collecting tar balls from the marshes.

News that the well may have already been blocked came as the state of Alabama announced that it is suing BP, and its partners on the Deepwater well, Transocean and Halliburton, for the "catastrophic harm" caused by the spill.

The attorney general Troy King declined to specify a figure for damages, telling Reuters: "We are suing them for the amount it will take to make Alabama whole."

There was also anger at a report in the Times-Picayune newspaper that BP had ordered claims adjusters to halve payments in August to those who lost business or income because of the spill.

The newspaper said it had obtained an internal BP email ordering the cuts. BP told the paper it was halving payments because the Obama administration was due to take over claims on 15 August. That takeover is now delayed.

The Obama administration's lead official for the oil spill, Thad Allen, said today that crews may have inadvertently sealed off any escape routes for oil when they pumped in mud and cement in the "static kill" operation earlier this month.

BP said it was reviewing data collected from four hours of testing on the well on Thursday before making its recommendations. Executives have said repeatedly in recent days that they saw no need to carry on drilling two relief wells.

But Allen had resisted until today when he told a briefing that the cement that entered the well from the top in the static kill may have sealed it off for good. "We may be the victims of our own success here. A bottom kill finishes this well. The question is whether it's already been done with the static kill," he said.

Allen warned that there was a small chance that cement pumped in during the static kill had travelled down to the reservoir and then back up in the outer casing, sealing the well from both ends.

Pumping more cement and mud in from the relief well could raise the pressure to dangerous levels, Allen said. But if BP and Allen do decide to go ahead, drilling on the relief wells will resume on Sunday, with the final kill complete in about four days.