Amid increasing anger over Cairo's handling of the killing of the 28-year-old Cambridge University graduate student, Egyptian detectives are now expected to hand over key evidence to their Italian counterparts on April 5.
If Cairo should fail to follow through, "I would urge our foreign ministry to urgently consider recalling our ambassador to Egypt for consultation," said Luigi Manconi, president of the human rights commission in the Senate.
Italy's Interior Minister Angelino Alfano on Sunday said Egypt agreed to extend the investigation after pressure from Rome, which had objected last week to Cairo insisting it had identified a criminal gang linked to Regeni's murder, after killing four gang members and finding the student's passport in one of the their apartments.
Egyptian police said they believed Regeni had fallen victim to the gang, which had hoped to force him to empty his bank-account.
Italian media and Western diplomatic sources in Cairo have voiced suspicions that Egyptian security services were behind the murder of Regeni, whose mutilated body was discovered nine days after he disappeared on January 25.
"I won't tell you what they had done to him," said Regeni's mother Paola. "I recognised him just by the tip of his nose. The rest of him was no longer Giulio".
She said she had taken a photograph of his battered body and was prepared to publish it if Cairo continued to refuse to share the findings of its probe with the Italian police.
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"What torments me is the thought that, before the first blow even fell, he knew that a door had closed forever. He had all the intelligence and culture to know what was about to happen to him," his mother said.
'All the world's ills'
"On Giulio's face I saw all the ills of the world. We have not faced such torture since the anti-Fascist era," she added.
Regeni had been researching labour movements in Egypt, a sensitive topic, and had written articles critical of the government under a pen name.
Prosecutors in Cairo on Saturday ordered the detention of four people over Regeni's murder, all of them closely related to the leader of the gang who was killed in the shootout with police.
Italy has so far rejected each of the vastly contradictory accounts Egypt has put forward, including allegations early on that the student had been working as a spy -- an accusation his mother furiously denied.
The family's lawyer, Alessandra Ballerini, said the investigators from Cairo "must bring us everything that's missing, including the phone records and data collected from the cell sites in the area and the security video footage from near the metro where he disappeared, and the area in which his body was found."
"We don't even know what Giulio was wearing when his body was discovered," she said, adding that the family wanted more information as well on the gang which had his passport, and how they could have acquired it.
Manconi said that if the data was not handed over, Italy's foreign ministry "should declare Egypt an unsafe country, which would without a doubt have a not insignificant effect on the numbers of tourists" there.