The body of 22-year-old Jessie Earl was found in undergrowth at Beachy Head near Eastbourne, East Sussex, in 1989, nine years after she disappeared from her bedsit in the resort.
A 1989 inquest into her death recorded an open verdict, but her now elderly parents John and Valerie Earl spent years trying to overturn its finding, believing their daughter was murdered.
In 2000, following a second police investigation dubbed Operation Silk, Sussex Police concluded that Ms Earl was murdered.
At the time, a coroner recorded an open verdict, despite the fact she was found naked and tied up with her own bra.
A cold case review carried out by Sussex Police in 2001 then ruled that she had been murdered, and last year, the High Court ruled that a fresh inquest should be held.
Scottish serial killer Peter Tobin has been linked to the murder but has refused ever to discuss any of his victims with detectives.
It had been suggested he could be called to give evidence via video-link but Tobin is gravely ill and rarely leaves his cell in Edinburgh’s Saughton Prison.
Tobin, 75, is serving life for raping and murdering Polish student Angelika Kluk, 23, and hiding her body in an underfloor storage the floor of a Glasgow church in 2006.
Tobin was also handed life terms for the murders of 15-year-old schoolgirl Vicky Hamilton, of Redding, near Falkirk, in 1991, and 18-year-old Dinah McNicol, from Chelmsford, the same year. He picked the student up as she hitchhiked home from a music festival.
Their bodies were found 17 years later, buried in the garden of his former home in Margate, Kent. Vicky Hamilton’s remains had been stored in Tobin’s former home in Bathgate – where he had abducted her as she waited for a bus home – and were cut in half.
He has been rushed to hospital for emergency treatment twice in recent months after falling ill in his cell.
Ms Earl’s family also want to see David Fuller investigated. Like Tobin, the man responsible for the bedsit murders in 1987 was living nearby when she went missing.