The 31-year-old Cuban, nicknamed 'La Sombra' ('The Shadow'), won light-heavyweight gold at Rio 2016 and dominated that weight class for years with his fourth consecutive world title coming in 2017.
Two years later, he was stunned by Kazakhstan's Bekzad Nurdauletov in the semi-finals and had to settle for bronze.
La Cruz has since moved up to heavyweight, although the cancellation of the Buenos Aires Americas Olympic qualifier means he could be forced to revert to light-heavyweight - and face a possible rematch with Nurdauletov - with his ranking points obtained in the lighter division.
Rio bronze medallist Erislandy Savon is currently Cuba's highest-ranked boxer at 91kg.
But if La Cruz was picked at his expense, he would bid to join an elite group of six boxers to win Olympic golds in two different weight classes.
Were La Cruz to succeed, he would become the heaviest boxer to do it with Hungary's three-time Olympic champion Laszlo Papp winning two light-middleweight titles after a first success at middleweight at London 1948.
"The key to victory is in the mentality, the technique and the tactics that we use. I will go to Tokyo with everything." - Julio Cesar La Cruz speaking to Cuba Debate
A rare setback for the Cuban stylist
La Cruz may have an orthodox stance, but that's about all that is orthodox about his boxing style.
With hands held low and his chin dangerously unprotected, the Camaguey native often makes his opponents look silly as he ducks shots and puts his dazzling hand speed to good effect.
He won his first Pan American Games title in 2011 but the man he beat in the final, Brazil's Yamaguchi Falcao, exacted revenge in the Olympic quarter-finals at London 2012.
After that came a period of almost complete dominance, with La Cruz twice defending his world title before beating Adilbek Niyazymbetov of Kazakhstan to win Cuba's first Olympic light-heavyweight gold in Rio.
Another world crown followed in 2017, and few would have bet against him making it five in Ekaterinburg.
He made smooth progress to the semi-finals where he came up against another Kazakh boxer, solid southpaw Bekzad Nurdauletov.
La Cruz's reputation did not faze his opponent one bit and, after a tight opening three minutes, Nurdauletov - who subsequently won last year's Asian Olympic Boxing qualifiers in commanding fashion - rocked him with a left early in round two and followed up with two more shots to bring about a standing eight count.
That made it a 10-8 round to the underdog and, while La Cruz rallied in the final round, he lost by a narrow split decision with four judges scoring it 28-28 and one giving it to Nurdauletov 29-27.
La Cruz remains unhappy with the decision saying, "The memories are not pleasant as I did not think I lost the fight and neither did my trainers. I recovered and finished much better than my opponent."
Why Julio Cesar La Cruz changed weight class
That defeat and constant struggles to make the weight prompted La Cruz, who has long been the captain of the Cuban boxing team, to move up from 81kg to the 91kg heavyweight division.
He told Cuba Debate, "It suits me better. I have developed in other areas that I could not before because I had to lose 12-13kg to get down to 81kg. I have incorporated strength and other exercises while retaining the same speed and explosivity."
The postponement of the Games actually helped La Cruz, who contracted chickenpox just before the Buenos Aires qualifier's original date of March last year.
Speaking to Adelante last year, he said, "I went to final training with my body almost covered in chickenpox welts. I would have taken the trip with the risk that the doctors would not let me compete."
It has also given him more time to adjust to his new weight class although a win over Rio 91kg silver medallist Vasilly Levit shows he is certainly on the right track.
For now, he is concentrating on this year's Games and winning a second gold.
He said, "I always train for gold. That is my goal, the spirit that our maestro Alcides Sagarra instilled in us.
"As a good Cuban boxer, in my mind there is no aspiration other than to be a champion. For this, I prepare daily because I feel capable of achieving it." - Julio Cesar La Cruz talking to Cuba Debate
He features in the Olympic Channel film The People's Fighters detailing Cuba's almost unparalleled success in boxing.