Kate needs to be here, Prince William says on latest trip

Kate, 42, has been the focus of speculation, rumour and bizarre conspiracy theories on social media leading to headlines and intrigue around the world during her absence from royal duties after she had abdominal surgery for a non-cancerous but unspecified condition in January.

On Monday, a video published by the Sun newspaper on its website showed the smiling Princess of Wales walking and carrying shopping bags alongside her husband William, the heir to the throne, at a farm shop in Windsor near their home.

Filmed by a member of the public on Saturday, it was the first footage of Kate taken since Christmas Day. Kensington Palace has not denied that the video is genuine, but has declined to comment on a matter it regards as being the royals' private time.

"If that video of Catherine sort of walking quite briskly with a shopping bag doesn't put an end to this frankly wild and baseless speculation, then I'm not sure what will," said Katie Nicholl, royal author and correspondent at Vanity Fair magazine.

Since being hospitalised in January, Kate's office has stuck to its official line that the princess was recovering well and that it would only provide significant updates, in keeping with the royal mantra of "never complain, never explain".

'JUST LEAVE THEM ALONE'

She is not expected to return to official duties until after Easter which falls on March 31, it said.

However, that has led to online rumours, and Kate's apology this month for editing a photo taken by William of her and her three children to mark Mother's Day only sparked further intrigue.

"A lot of people out there didn't seem to want to believe that the simple truth of the matter was that she was at home recuperating from a clearly serious abdominal surgery," Nicholl told Reuters.

On Tuesday, William visited Sheffield in northern England to unveil new investment in his scheme to tackle homelessness, one of his main campaigning themes.

Looking relaxed, he laughed and joked with staff and volunteers, and later referred to Kate during discussions about homelessness and early childhood, one of his wife's key projects.

"We're venturing into my wife's territory," he said. "She needs to be sat here to hear this."

He made no reference to the ongoing speculation about his wife, and looked relaxed as

Kate's illness also comes at the same time King Charles is undergoing treatment for an unspecified form of cancer, meaning he too has been forced to postpone public engagements.

Such has been the febrile gossip about the royals that the British embassies in Ukraine and Moscow even posted statements on X on Monday to clarify that fabricated reports on social media that the king had died were indeed fake.

Charles has generally maintained a more visible profile as he keeps up with official state duties while undergoing treatment, and was pictured on Tuesday meeting veterans from the Korean War at Buckingham Palace.

While the online world remains awash with rumour, outside the palace, passers-by said Kate and Charles should be allowed to recover in peace.

"I think that people should just leave them to get on with their lives," said Judy Bradley, 61. "It's nice to see her, I'm glad she's in good health, hopefully everything is well, but I think, just leave them alone."