Firms involved in the refurbishment of Grenfell Tower have asked the public inquiry into the fire for a guarantee that anything they say in the hearings will not be used for any prosecution.
They want a guarantee from Attorney General Geoffrey Cox that they will be protected when they give evidence.
The fire in June 2017 killed 72 people.
The Royal Borough of Kensington and Chelsea has "apologised unreservedly" for a "number of failings" by its building control services.
The inquiry's second phase, which began on Monday, is looking at how the building came to be covered in flammable cladding during its refurbishment between 2012 and 2016.
Experts have previously said the work failed to meet building regulations.
Representatives from organisations including cladding company Harley Facades, building contractor Rydon and the Kensington and Chelsea Tenant Management Organisation made the application for the guarantee.
The request was read out by inquiry chairman Sir Martin Moore-Bick at the hearing in London.
It was met with groans from survivors and families of the victims in the room, who are likely to strongly oppose the move.
Michael Mansfield QC, who is representing victims, told the inquiry the timing of the application was "highly questionable and highly reprehensible".
"There has been plenty of time for this to be considered," he said.
"It has caused immense anxiety, distress and anger at a time which has come throughout a much longer period of waiting after this disaster, of waiting to get to the point of accountability, as it were, to be almost thwarted at the doors of the court."
Sir Martin said he would hear the firms' application on Thursday afternoon.
Scotland Yard is conducting its own investigation into gross negligence manslaughter, corporate manslaughter and health and safety offences and said in September it had so far completed 17 interviews.
Under the Inquiries Act 2005, people giving evidence at an inquiry have the right to withhold information which might incriminate them.
But the attorney general has the power to rule that no evidence given by witnesses will be used against them in criminal proceedings, except if they are charged with conspiring to or giving false evidence to the inquiry.
The first part of the Grenfell Inquiry examined events that took place on the night.
It found the cladding was the "principal" reason for the rapid and "profoundly shocking" spread of the fire at the 25-storey building.
It also concluded that "many more lives" could have been saved if the advice to residents to "stay put" had been abandoned earlier.
At the opening of the second phase on Monday, the inquiry heard that - with the "sole exception" of the Royal Borough of Kensington and Chelsea (RBKC), which accepted that the refurbishment should not have been signed off - all organisations involved in the work had denied responsibility for the fire in "carefully crafted statements".
The following day, emails disclosed to the inquiry suggested that companies knew a planned cladding system would fail in the event of a fire.
On Wednesday, RBKC counsel James Maxwell Scott QC told the inquiry: "Grenfell Tower is a tragedy which should never have happened. This council could have done and should have done more to stop it happening."
Among the failings by its building control services during the tower's refurbishment, Mr Maxwell Scott said officials failed to ask for comprehensive details on the cladding system involved in the refurbishment and failed to identify the insulation materials used were not of limited combustibility.
Survivors group Grenfell United said there was "no true remorse" in the admission by RBKC, adding it was "insulting to survivors and bereaved families for them to suggest they are being honest about their role in our suffering".
RBKC owned the west London block but it was run by an arms-length body, the now-defunct Kensington and Chelsea Tenant Management Organisation (KCTMO), which oversaw the refurbishment.
The inquiry has heard that the cladding was chosen to help save money on the refurbishment.
However, Alice Jarratt of the KCTMO told the inquiry on Wednesday it was "not the case" that minimising costs was its "only consideration for the project".