I'm not sure if this conjecture is less hard than Goldbachs conjecture:
any integer greater than $2$ is the sum of an odd prime and two squares of integers.
Every prime of the form $4n+1$ is the sum of two squares.
Every natural number is the sum of four squares
may or may not be helpful.
I've tested the conjecture for all integers less than $10^6$.
Even if the conjecture maybe is to hard to prove I would like to see ideas and heuristics about it. Or counter-examples!