Weak odd Goldbach conjecture for sums of consecutive progression terms

Let $A(p,q)$ be the arithmetic progression whose first two terms are the odd primes $p$ and $q$, and let $S_m(p,q)$ denote the sum of its first $m$ terms. For each positive integer $n$,

$$S_{2n+1}(p,q)=(2n+1)(nq-(n-1)p).$$

Weak odd Goldbach conjecture. For each odd positive integer $a>2$, there exist a positive integer $n$ and odd primes $p$ and $q$ such that

$$a=S_{2n+1}(p,q)=(2n+1)(nq-(n-1)p).$$

This is presented as the odd analogue of the weak even Goldbach conjecture, extending representations by two or more initial terms of prime-started arithmetic progressions. The source gives no proof or resolution.