## Write $2n+1>14$ as $p+q+r$ with $p,q,r$ odd primes and $2p+4q+6r$ a square

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?