The parents of Giulio Regeni, the Italian student murdered in Cairo three years ago, have written to the Egyptian president demanding he extradite five men to Italy to face trial.

Paola and Claudio Regeni published their emotional plea in an open letter to Abdel Fatah al-Sisi. They condemned what they see as false promises by Egyptian authorities to investigate the murder of their son, and a cover-up.

“As long as this barbarism remains unpunished, until all those who are guilty, regardless of their position, are brought to justice in Italy, no one in the world can stay in your country and feel safe,” the letter reads.

Facebook Twitter Pinterest A protest in Rome held a year after Regeni’s disappearance. Photograph: Andreas Solaro/AFP/Getty Images

Regeni’s mutilated body was found by the side of an outlying Cairo desert road in February 2016, bearing clear signs of torture, following his disappearance on 25 January that year. The 28-year-old’s mother, Paola, later stated she only recognised that the corpse belonged to her son by “the tip of his nose”.

Letters were carved into the side of Regeni’s body by one of his torturers and there was evidence that Regeni had been followed and investigated by the Egyptian security services prior to his death. Nevertheless Egyptian authorities thwarted Italy’s efforts to investigate the murder, even as Sisi himself promised the Regeni family “the truth”.

Egyptian officials offered alternative theories, suggesting the young Italian had been murdered by Islamists, or claiming that a gang of five men later killed in a shootout with Egyptian police were responsible for his death. While Egypt later backed away from these theories, authorities in Cairo continued to resist efforts to name suspects or even provide the full range of evidence demanded by prosecutors in Rome.

Last November, frustrated by a clear lack of progress in the investigation, Italian officials named five members of Egypt’s powerful National Security Agency as suspects. Egypt’s only response came from the country’s State Information Service, which claimed: “Egyptian law does not recognise what is called ‘the record of suspects’.”

Earlier this month, according to the Regeni family lawyer, Alessandra Ballerini, Italian officials requested fresh evidence from their Egyptian count erparts after a new witness came forward, claiming that one of the five men accused of complicity in Regeni’s disappearance had been overheard discussing the kidnapping at an African security conference in 2017. According to the unnamed source quoted in the Italian daily Corriere della Sera, the suspect was overheard saying: “We thought he was an English spy. After putting him in a car we had to beat him. I hit him in the face.”

The letter from Regeni’s parents adds fresh pressure on Egypt to extradite the five security officials named last year.

“It is hard to believe that the people who kidnapped, tortured and killed our son Giulio, who lied and dug up dirt on him, organised countless misdirections, murdered five innocent men blamed for his death, that all these people acted without you knowing or against your will,” the parents write.

Public anger in Italy at Regeni’s murder has not dimmed, even as some politicians have slowly repaired ties with Egypt.

In a speech on 29 April, Massimo Ungaro, an MEP with Italy’s Democratic party, said his country was “getting played by Egypt”.

Riccardo Noury of Amnesty International Italy said “there was no easy way out” for Sisi and that he was duty bound to respond to the letter.

A spokesman for the Egyptian presidency did not respond when contacted by the Guardian for comment.