Sergei Trifonov, however, believes he has solved the riddle, and that the treasure - ornately carved panels of glowing amber, formed from fossilised resin - lies underneath a bunker in the Russian enclave of Kaliningrad.
"Believe me or not, it's there, 12 metres down in the sub-soil," he said, pointing to the entrance of a bunker that sheltered the Nazi high command in the last hours of the Battle of Königsberg.
"This place was built (in February 1945) with two aims: accommodating the headquarters of General Otto Lasch and storing the treasures of Königsberg, a city under siege," the historian turned journalist and lecturer argued.
Königsberg, in what was then German East Prussia, is now Kaliningrad, the capital of Russia's westernmost region of the same name. The Nazis removed the treasure from a palace that once belonged to empress Catherine the Great outside Saint Petersburg after invading the Soviet Union in 1941.
Once hailed as the eighth wonder of the world, the trophy was brought here and stored in the former castle of the Teutonic Knights in the centre of the city. But its subsequent fate remains unknown amid the turmoil of war and heavy bombardment of the city by the Allies.
The Red Army seized the ruined city on 9 April, 1945. Stalin triumphantly annexed it and renamed it Kaliningrad, after a leader of the Supreme Soviet. The Amber Room - with its 35 square metres of panelling - was sighted in the last month of the final attack, after it had been disassembled and stored in cases. After that, it vanished.
There are many theories about what happened next. The panels may have been destroyed by bombs, secretly moved to Germany or hidden in the maze of tunnels under Kaliningrad.
To test his theory, Trifonov has begun to probe the soil under the bunker using a ground-penetrating radar, and pump out water. He has already unearthed a brick-lined room.
The bunker is 300 metres from the site of the castle - demolished in 1967
- that sheltered the Amber Room. Its iron gate features Viking symbols and a Teutonic cross, suggesting it wasn't only for military use, Trifonov argues. "I'm sure that the Amber Room wasn't taken away. The theory that it is now in Germany doesn't make sense," he said. "If that was so, the Germans would have found it."
Working alone with help from several sponsors, Trifonov says his supporters include Kaliningrad's governor. Nevertheless, not everyone takes his theories seriously.
"He's a good storyteller who can't prove anything," said Vladimir Kulakov, an expert at Russia's Institute of Archaeology, who has also dug in the soil under the bunker in the search for the Amber Room.
Anatoly Valuyev, deputy director of Kaliningrad's History and Art Museum, which takes in the bunker, was more circumspect.
"It's good that people think that the treasure is there. They have energy and the museum gains from this," he said, citing the water being pumped out. "We still hope that the (Amber) Room is somewhere in Kaliningrad," he said. "There are plenty of underground sites left to explore. If they don't find it here, they'll look elsewhere."
A reconstruction of the Amber Room was opened at the Catherine Palace near Saint Petersburg in 2003.
If the Amber Room is indeed in Kaliningrad, it may not be far from home. This region, on the coast of the Baltic Sea, is the world centre for amber extraction. Its vast deposit represents 90 percent of global supplies of the sought-after material.
Every year, the state-owned amber factory in the village of Yantarny, whose name means Amber in Russian, extracts 300 to 350 tonnes of amber in open-cast mining. The most attractive or unusual specimens can hit a price of €230 ($330) per kilo.
Most of the amber is then exported in raw form, usually to Poland, where it is polished and turned into jewellery.