Ms McKay, the wife of newspaper executive Alick McKay, was kidnapped for a £1m ransom in 1969.
But it was a case of mistaken identity, as her killers had intended to seize and ransom Anna Murdoch, the wife of Alick’s then boss, Rupert Murdoch.
Two brothers, Arthur and Nizamodeen Hosein, were convicted of Mrs McKay’s murder and kidnapping 55 years ago, though her remains were never found.
Hoping to find her body, two of her children – Ian McKay and Dianne Levinson – asked the High Court to allow a “ground-penetrating radar survey” at two neighbouring London properties.
On Tuesday, the court denied them the injunction that would have enabled the survey.
Giving his ruling, Justice Richard Smith said that while Ms McKay was the victim of “abhorrent crime”, he had to consider the case “objectively and dispassionately”.
He said: “I was not persuaded that even if a survey was carried out, that it would be conclusive one way or the other, that it would produce incontrovertible data.”
He added: “The evidence of the presence of Muriel McKay’s remains at the premises, such as it is presently, seems thin.”
It comes after new information from Hayley Frais, whose father ran a tailor shop on the site in Bethnal Green Road, where Arthur Hosein worked at the time of the killing.
She claimed her father said on his deathbed that he noticed a strong smell at the premises after Ms McKay disappeared.
Benjamin Wood, speaking for Ms McKay’s children, admitted Frais’s testimony did not cross the “evidential threshold” for police to survey the site.
One of the affected homeowners, Madeleine Higson, opposed the injunction bid.
Callum Reid-Hutchings, speaking for her, argued that the survey would constitute a “significant intrusion into her private space and her right to peaceful enjoyment of her home”.
He added that his client had “considerable sympathy” for the family of the deceased, but the injunction lacked a “proper legal foundation”.
It was “telling”, he added, that police had decided not to scan the garden.
Dianne McKay, Muriel McKay’s daughter, told The UK Tonight with Sarah-Jane Mee the decision left her “very disappointed” but said the family will “continue with the legal process”, adding: “We’re not giving up just because the judge says we can’t go in.”
Muriel McKay’s grandson, Mark Dyer, said they believe “in the legal process, that we will prevail and we will get into the property to do the scan eventually… all we want is an end to this life sentence for my mother and her siblings”.
The court also heard that homeowner Ms Higson had endured “borderline harassment” and “significant distress” by the McKay family.
This included repeated attempts to gain access to the property through false representations.
Read more:
Police call off search for remains of Muriel McKay 55 years
The judge said he would also have refused the injunction on the basis of this “egregious conduct”, which included “threats, deception, dishonesty, lies, bullying and harassment”.
He said: “It was obviously immoral and, in part at least, likely unlawful. There was no justification for it.
“It seems to me in their desperation to find an answer to what has happened to Muriel’s body, the claimants and Mr Mark Dyer have lost a sense of perspective and also respect for the interests, concerns and safety of others who are perceived to stand in the way of their campaign.”
