The weak Goldbach conjecture states that any odd number $ 2n+1>5$ can be written as the sum of three primes. This was finally confirmed by Helfgott in 2013.
Motivated by my 1-3-5 conjecture (cf. http://oeis.org/abs/A271518), I formulated the following conjecture in 2016, which refines the weak Goldbach conjecture.
Conjecture. For any integer $ n>6$ , we can write $ 2n+1=p+q+r$ , where $ p,q,r$ are odd primes with $ 2p+4q+6r$ an integer square.
I have verified this for $ n$ up to $ 10^5$ . For example, $ 2\times 13+1=7+17+3$ with $ 7,17,3$ odd primes and $ $ 2\times 7+4\times17+6\times 3=10^2.$ $
QUESTION. Is the conjecture true? If true, how to prove it?