The most ironic thing about some of the misconstrued history is that lies told to cover up the fact that the family was dead ended up fueling rumors of their escape! For example, the Perm stories. These stories were totally false and most likely planted by Bolsheviks to cover the fact that Alexandra and the girls had been killed, since they had just made peace with Germany, and the Germans had demanded the 'princesses of German blood' (Alix and her daughters) to be handed over safely to them. So, this was not very good for relations to find out they had been brutally killed, was it? No one immediately knew the entire family had perished. Even the respectable NY Times printed a story it believed to be true, saying only the Tsar had been shot, and Alix and the children had been moved to an undisclosed location. So there were lies told, but it wasn't because they were alive, it was because they were dead!
In Felix Yussoupov's memoirs, he tells how he, his family, the Dowager Empress and many other royals and nobles had been holed up in the Crimea in the summer of 1918, hearing rumors and rumors of rumors that the family was dead, alive, and back and forth until they didn't believe any of it. Also, there is a precedence for this lying to cover murders: the main executioner of GD Ella, the KR sons, Prince Paley and Sergei Mikhailovich the day after the IF wrote in his memoirs that after tossing them down the mine shaft, he and his men returned to the town, rang a bell and announced the prisoners had been taken away by 'unknown persons.' This diverted the blame from them, but gave false hope to family members, who believed they were still alive. However, no one had to wait long to learn the truth. In little over a week, the White army took the town and the bodies were found in the pit, so all rumors of survival and escape were put to rest. Sadly, since the Romanov bones were not found for many decades, lots of rumors, tall tales, speculation and incorrect information was allowed to spread, perpetuate and manifest in ways it never should have. Some people believe some of it to this day, because the mystery is more fun and exciting than the truth- that they all died horribly that night in 1918.
It's true that two bodies remain missing, and until we find them, there will always be wondering about what happened to them and if they were alive. Considering the brutality of the murders and the condition of the bodies, their survival is most unlikely. There are reports that two of the bodies were burned along the way, and this would account for the missing ones. Sadly, we may never find these remains to prove it, so the guessing game will go on. We do know that none of the claimants were who they said they were, not even Anna Anderson, whose claim was proven false by DNA testing. It would be nice to believe that Anastasia and Alexei got away and lived long lives in obscurity, but it's just not realistic.