Katherine (left) and Sheila Lyon have been missing since 1975. Sheila Lyon was 12, her sister Katherine 10, when they vanished.
Maryland (CNN) -- The two little girls vanished nearly 40 years ago, their disappearance long faded from the public consciousness, their names etched in a stone marker their hope-sapped parents placed in a local cemetery.
But Maryland police have been tenacious about solving this cold case. And on Tuesday they will breathe new life into it, when they announce a possible breakthrough.
A convicted sex offender, who is currently in jail, was near Sheila and Katherine Lyon on that March day in 1975, when they went missing.
Officers will release the offender's name and photo at around noon Eastern and ask the public if they know what he's done in the past and the places he's been. But Montgomery County police have stopped short of calling him a suspect.
The public eye lost sight of the case years ago. If they're alive, the women are around the age of 50.
After years of holding out hope, their parents gave up.
Sixteen years ago, they set a marker for them in a cemetery with their dates of birth and the date they went missing, the Washington Post reported.
But the police department has not forgotten Sheila, who was 12, and Katherine, who was 10, the day their older brother, Jay Lyon, saw them last.
Maybe, in part, that's because he joined the police force as a homicide detective just a few years after that.
Walk to pizza parlor
The girls must have been full of anticipation, when they decided to step out for some fun in their neighborhood not far from Washington, D.C.
Their birthday dates were just one day apart, and celebrations were coming up in just a few days. It was also the first day of Easter school vacation.
They walked half a mile to a nearby strip mall to check out the Easter decorations, the Montgomery Gazette reported.
Their mother told them to be home by 4 p.m. At 7 p.m. police received a call to report them missing, the privately run missing persons website CharleyProject.org reported.
Police swarmed out looking for them and questioned passersby.
A medley of witnesses said they'd seen the girls. One saw them walking to the shopping center.
Another -- a child -- saw them talking to a stranger, a man holding a microphone and a briefcase. Police drew a composite of him based on the witness's description.
But the girls walked away from him.
An hour later, their brother Jay, 15 at the time, spotted them at the shopping center eating pizza. And after that, someone else saw them walking in the direction of their home, according to the CharleyProject description.
Had they arrived, they would have made it home on time.
Back then, it was common for children to walk to places in the neighborhood without their parents. But when the girls disappeared, that feeling of safety was shattered, the Gazette wrote.
The girls' father, John Lyon, was a well-known radio host in Bethesda at the time, so the story of his daughters' disappearance made the rounds.
Anonymous callers rang the family demanding money, CharleyProject.org claims. One of them demanded their father deposit $10,000 in a particular location.
He complied. No one picked up the cash. The man called again, but when asked for evidence that he actually had the girls, he broke off contact.
In the ensuing years, the tips faded. The case languished.