Friedlander–Goldston–Iwaniec–Suriajaya conjecture for the weighted Goldbach function

Define the weighted Goldbach representation function

$$G(n)=\sum_{\substack{m_1+m_2=n\\2\nmid m_1m_2}}\Lambda(m_1)\Lambda(m_2)$$

and the singular series

$$\mathfrak{S}(n)=2C_2\prod_{\substack{p\mid n\p>2}}\left(1+\frac{1}{p-2}\right),$$

where $C_2$ is the twin prime constant. Friedlander–Goldston–Iwaniec–Suriajaya conjecture. There exists a fixed constant $\delta$ with $0<\delta<1$ such that, for all sufficiently large even $n$,

$$\delta\mathfrak{S}(n)n<G(n)<(2-\delta)\mathfrak{S}(n)n.$$

The source describes this as a weak form of the Hardy–Littlewood conjecture sufficient to rule out exceptional zeros, but does not establish the assertion.