The First Hardy-Littlewood Conjecture (k-tuple Conjecture) (can be found here) is usually stated without a precise error term.
My question is, is there any precise big-$O$ error term that's conjectured? For example, suggested by the integral $\int_2^{x}\frac{dt}{\ln^k(t)}$, is it $$C\frac{x}{\ln^k(x)}+O\!\left(\frac{x}{\ln^{k+1}(x)}\right)$$ for the proper constant $C$?
I know this
$\displaystyle \pi(x,\mathcal{H})=\mathfrak{G}\int_2^x \frac{dt}{\log(t)^{|\mathcal{H}|}}+O(x^{1/2+\epsilon})$