Unfortunately, they award it on the 4n+2 years. As someone born on a 4n+0 year, I’ll have just 38 years, which is too severe a disadvantage for me to stomach, so I didn’t bother with it.
4n+2 people, you have no excuses.
So your hypothesis does seem to bear out: people born on 4n+0 years are at a significant disadvantage for winning a Fields medal.
The theorem went unproved for 385 years until Andrew Wiles eventually proved it in 1995, winning the Abel prize (because he was too old to win a Fields medal). Most people think Fermat didn't have a proof - just an intuition - as Wiles' proof uses a bunch of extremely sophisticated and powerful results that came well after Fermat.
The GP's comment is particularly clever because until Wiles, the Riemann hypothesis and Fermat's last theorem were the two most famous unsolved problems in maths, so were often talked about together as being these almost unassailable challenges.