The prime-divisor positivity implication for Goldbach convolution sums

Let $N$ be an integer, let $p$ range over the primes dividing $N$, and let $\Lambda_0$ denote the function used in the source. Prime-divisor positivity conjecture. If

$$\sum_{p\mid N}\sum_{n\leq \frac{N}{p}}\Lambda_0(n)\Lambda_0(N/p-n)>0,$$

then

$$\sum_{n\leq \frac{N}{p}}\Lambda_0(n)\Lambda_0(N/p-n)>0$$

for every prime $p$ dividing $N$. The text proposes this implication as the statement through which the Goldbach conjecture will be attacked, but supplies no resolution.