Uniform weak error conjecture for prime-pair counts

For $d>0$, define

$$I(x,d):=\int_2^{x-d}\frac{dt}{\log t\,\log(t+d)},\qquad \widetilde{G}(x,d):=\mathfrak{S}(d)I(x,d),$$

and write $G(x,d)=\widetilde{G}(x,d)+E(x,d)$. The uniform weak error conjecture for prime-pair counts. The variable-difference Hardy–Littlewood formula holds with

$$E(x,d)=o\left(\frac{x}{(\log x)^4}\right)$$

uniformly for

$$2\le d\le \frac89x.$$

This weaker conjecture is the estimate the paper needs for its analysis of prime difference champions. It is presented as sufficient for the stated problems, while the stronger square-root error conjecture above would imply substantially more.