IIRC, there was an account about a possible Alexei in
"The Last Tsar : The Life and Death of Nicholas II" -- by Edvard Radzinsky
This author agrees that two of the Romanovs may have escaped; he spoke with many of those involved in the execution, or with their family members.
However, he traces a possible Alexei to St. Petersburg, to an asylum where he was placed for his protection, and died
shortly after World War II.
So, while there is a considerable possibility that two of the Romanovs survived, there is more than one candidate
for Alexei.
(BTW, I'm going on memory here; I THINK that is the book
which delves into the mystery of Alexei, though I may
have mis-remembered. )
BTW, also, if you need a DNA test, you can get some from the Mountbatten family (British royal family tree), since Prince Philip is related to some Romanovs.
[However, he traces a possible Alexei to St. Petersburg, to an asylum where he was placed for his protection, and died shortly after World War II.]
I am sure there are a number of candidates. I am well aware that a "secret oral tradition" in a Soviet family could be used as an emotional survival mechanism to get through communism...regardless of the truth of the tradition.