Prince Harry and Prince William are set to reunite again this week at a ceremony honoring their late mother Princess Diana

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Prince William and Prince Harry's reunion at grandfather Prince Philip's funeral did not put an end to their rift.

In a new article for the Daily Mail, Robert Lacey -- author of Battle of Brothers and consultant to Netflix's The Crown -- wrote that though some hoped the funeral "would bring the warring brothers together in an atmosphere of reflection," the result was not as positive. 

Although the pair shared a brief public chat outside St. George's Chapel that day, an argument broke out between William and Harry "within minutes of the siblings getting inside the castle and beyond camera vision," according to the royal historian.

"They started quarrelling again," he added.

"There they were, at each other's throats as fiercely as ever," a friend of the family told Lacey. "The rage and anger between those two has grown so incredibly deep. Too many harsh and wounding things have been said." 

Prince William, Duke of Cambridge and Prince Harry, Duke of Sussex
Prince William and Prince Harry at Prince Philip's funeral
| Credit: Gareth Fuller/WPA Pool/Getty Images

Lacey went on to write that the brothers did not have a family talk with Prince Charles.  

"There was no reconciliation, and no brotherly sit-down or 'mini summit' following Prince Philip's funeral on April 17," Lacey wrote, later adding, "The conflict between Diana's two bitterly divided sons does not seem likely to end any time soon."

RELATED: Prince Harry and Prince William's Fallout Has Left Prince Charles 'Shellshocked,' Says Source

As PEOPLE previously reported, the schism between the brothers began when William expressed concern about the speed at which Harry's relationship with then-girlfriend Meghan Markle was moving prior to their May 2018 wedding.

According to new chapters in Lacey's book, things worsened again in October 2018 when William learned of allegations from staffers who said they had been bullied. (When the bullying claims first surfaced in The Times in March, the Duchess of Sussex's office strongly denied the allegations and said they were part of a smear campaign against the Duchess, 39. The palace set up an independent inquiry, which is still "ongoing," a source confirmed to PEOPLE.)

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The Funeral Of Prince Philip, Duke Of Edinburgh Is Held In Windsor
Prince William and Prince Harry with Peter Phillips at Prince Philip's funeral.
| Credit: Alastair Grant/WPA Pool/Getty Images

Reconciliation between William, 39, and Harry, 36, has been made more difficult by geographic distance and the challenges of the pandemic.

"Everyone was optimistic," a source close to the royal household told PEOPLE earlier this week of the pair's reunion at Philip's funeral, "but nothing really happened."

Added another close source: "There was not really any time to build bridges."

There is, however, another reunion just around the corner: on July 1, both brothers are expected to attend the unveiling of the long-awaited statue of their mother, Princess Diana, at Kensington Palace Gardens, a project they commissioned together several years ago.