The fate of the three inmates who successfully escaped from San Francisco’s Alcatraz prison in 1962 is one of America’s greatest unsolved mysteries.

Their bodies were never found and the manhunt is still on 54 years later. But could a trio of common crooks really have survived paddling across the frigid, shark-infested waters surrounding The Rock on a makeshift raft cobbled together from raincoats?

A newly surfaced letter has reignited hope that the so-crazy-it-just-might-work scheme — made legendary by the Clint Eastwood film “Escape from Alcatraz” — wasn’t just a success, but that one of the men may still be alive today.

“My name is John Anglin. I escaped from Alcatraz in June 1962 with my brother Clarence and Frank Morris. I’m 83 years old and in bad shape. I have cancer,” reads the letter sent to the San Francisco Police Department in 2013 and aired for the first time by a local CBS affiliate Wednesday.

“Yes we all made it that night but barely!”

The letter says John — who would now be 86 — lived on the lam for years in Seattle, North Dakota and now Southern California. Morris died in 2008 and Clarence in 2011, it claims.

“John” says he’s reaching out to the feds because he needs a doctor to treat his cancer.

“If you announce on TV that I will be promised to first go to jail for no more than a year and get medical attention, I will write back to let you know exactly where I am. This is no joke.”

The US Marshals — which has kept the case open for decades — says the FBI performed a forensic handwriting analysis on the letter, but the results were “inconclusive.”

Family members tell The Post it doesn’t look like his handwriting — but they do believe all three men survived the breakout and that John is still out there.

“The handwriting don’t look anything like it. But that’s not to say John didn’t have somebody else write the letter. They’re pretty intelligent, they know the deal with fingerprints,” says John’s nephew David Widner, 51.

The Marshals say the letter yielded no leads, but the case remains open — even if the agency doesn’t seem thrilled about the possibility that three thieves have evaded their clutches for more than five decades.

“There is absolutely no reason to believe that any of them would have changed their lifestyle and became completely law abiding citizens after this escape,” the agency huffed in a statement to CBS.

But experts say credible new evidence continues to emerge suggesting that the men survived.

“I definitely believe, with a lot of new evidence that has come to light, that it’s possible,” says Michael Esslinger, a historical researcher who has penned several books on the escape and has been studying it for decades.

“For the first 20 years I was doing research, I was very adamant they had died. In the last seven or eight years, I really changed my opinion.”

It’s remarkable enough that the trio of small-time bank robbers made it off the famously inescapable island at all — let alone survived the treacherous swim to shore.

Morris was a career criminal from DC who spent his childhood in and out of foster homes and had already racked up a lengthy rap sheet in his teens.

The Anglin brothers were poor farmers from Florida who had turned to sticking up banks with toy guns.

Morris was a natural brainiac, according to Esslinger, while the Anglins had learned to be very crafty.

“They grew up with 14 brothers and sisters. They didn’t have a lot, so they had to make do with what they had,” says Widner.

“My mom said they were like MacGyvers.”

Plus, all three were seasoned escape artists, and all had ended up on Alcatraz precisely because they busted out of other joints.

“Escape from Alcatraz” depicts Eastwood’s Morris as the brains behind the scheme, but Esslinger says it was more of a team effort among the three — plus a fourth, car thief Allen West, who ultimately failed to escape.

The plan otherwise played out roughly the same as it does in the 1979 film: Over six months, the men quietly carved out the ventilation ducts in their cells using spoons and a drill made from a vacuum cleaner motor until they could access a utility corridor behind the walls.

There, they created a workshop and constructed a raft and life jackets out of pilfered rain jackets — using steam pipes to seal the seams — and made paddles from scrap wood.

They also made models of their own heads out of soap, toilet paper and real hair taken from the prison barbershop, to use as decoys in their beds on the night of the big break.

Finally, on June 11, 1962, the three slipped out — West couldn’t get through his duct and was left behind — and climbed to the prison roof, slid down a smokestack, scaled a fence, inflated their raft with a concertina and set sail for nearby Angel Island.

Thanks to the realistic fake heads, the guards didn’t realize what had happened until the next morning.

“If they made it to the mainland and stole a car, they could’ve been in Mexico by the time the first bell went off,” says Esslinger.

A massive manhunt was mounted but it failed to find much trace of the fugitives — dead or alive — other than a paddle and some scraps of life jacket.

Alcatraz officials believed they couldn’t have possibly survived the cold, choppy waters, but Widner says the family has always been sure they made it — and new clues over the past decade have only made them more sure.

“Our family has always believed that they made it,” he said, noting that the brothers grew up swimming in the icy waters of Tampa Bay in winter.

“We have a lot inside the family that we haven’t shared.”

In 2010, another Anglin brother, Robert, made a deathbed confession to his kin: He’d been in touch with John and Clarence several times in the first 25 years they were on the run, the family says.

Four years later, Dutch scientists took up the cause and used 3-D modeling of the tides at the time to prove that it was possible they survived — so long as they started paddling between 11 p.m. and midnight.

“In the worst case the paddling of the escapees is futile . . . the escapees are either swept into the Pacific if they entered the water before [11 p.m.], or they are pushed deep into the bay and discovered if they enter the water after midnight,” wrote researcher Rolf Hut.

“In the best case, the escapees peddle northwards with a speed of almost 1 k.m. per hour, an almost olympian effort. In that scenario, they most likely survive and make it to the north side of the Golden Gate bridge.”

In 2015, the History Channel filmed a documentary about the case — with the participation of the family, Esslinger and the Marshals Service — that yielded several big clues.

The family allowed investigators to dig up the remains of another brother, Alfred, to compare them with some bones that washed ashore in 1983 and authorities had believed could be one of the escapees.

The DNA test proved it wasn’t an Anglin — shattering what many thought they knew about the case, Esslinger says, including him.

“I always thought that was a powerful piece of evidence,” he said.

The Anglins also revealed during the show that in the 1990s, a drug smuggler and family friend told them he’d run into John and Clarence at a bar in Brazil in the 1970s — and gave them a photo apparently showing the two as middle-aged men, at a farm they’d bought down there.

The Marshals don’t think it’s legitimate, but a forensic expert on the documentary found it was “highly likely” that they’re the same men.

“When you work these types of cases, there’s a feeling you get when stuff starts to fall into place,” Art Roderick, the retired US Marshal who was lead investigator on the case for 20 years and who worked on the documentary, told The Post at the time.

“I’m getting this feeling now.”

But looking back on the show now, Widner is livid that investigators he worked with never told the family about the 2013 letter — and if his uncle was really out there dying from cancer, he hopes it isn’t too late.

“It’s inhumane to not let the family know they received a letter, whether it was him or not,” he said.

“At least we would have known it could possibly be John reaching out. That upset my mom a lot … we would love to get the word to him that if he would get in touch with us, maybe we could help him.”