I was curious what were the best equidistribution estimates toward the Elliott-Halberstam conjecture $$\sum_{q\le Q}\max\left|\Theta(x;q,a)-\frac{x}{\phi(q)}\right|<<\frac{x}{(log x)^A}$$ with $Q<x^{1/2+\eta}$, $\Theta(x;q,a)=\sum_{p=a\ (\rm{mod}\ q), p\le x} \log p$.
Zhang proved famous improvements upon Bombieri-Vinogradov 2 years ago. Then Polymath improved on him. The papers are quite long and difficult for me to comprehend well. Many estimates are for modifications of the Elliott-Halberstam average errors above so it's not clear what the best known $\eta$ in the above inequality is.
I would like to know even more whether there are hopes to still improve on those bounds. Are there clear qualitative reasons why they cannot reach the conjectured exponents -arbitrarily close to 1, any $\eta<1/2$? I understand they rely on bounds for individual (hyper)Kloosterman sums. Would the Linnik conjecture on sums of Kloosterman sums help obtain better results?
Thank you.