I've checked here for discussions of Helfgott's proof of the weak GC and found nothing that helps me with the following; apologies if I missed something.
I'm probably being naive here (please enlighten me), but I’m thinking that there is a natural conjecture that is intermediate between the strong and weak Goldbach conjectures. The strong GC can be stated as the weak GC (in its "three odd primes" formulation) plus the additional constraint that the smallest of the three primes can always be 3.
My question is: is there a larger number $N$ for which it can already be stated that the smallest of the three primes referred to in the weak GC can always be at most $N$? If not, what can be said about this question?