Weak even 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 $n$,

$$S_{2n}(p,q)=n((2n-1)q-(2n-3)p).$$

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

$$a=S_{2n}(p,q)=n((2n-1)q-(2n-3)p).$$

This extends the two-term formulation of even Goldbach's conjecture to sums of an arbitrary positive even number of initial terms of such arithmetic progressions. The source reports heuristic and computational motivation but no proof.