For $a$ and $q$ positive integers such that $a\lt q$ and $(a,q)=1$, let $\pi(x;q,a)$ be the number of primes $p\equiv a\pmod q$ below $x$. One can show that $\pi(x;q,a)\sim \dfrac{\pi(x)}{\varphi(q)}$ where $\pi(x)$ is the number of primes below $x$ and $\varphi(n)$ the number of positive integers $k$ less than $n$ such that $(k,n)=1$. Let's denote by $E(x,q)$ the quantity defined by $E(x,q):=\displaystyle{\max_{(a,q)=1}\vert\pi(x;q,a)-\dfrac{\pi(x)}{\varphi(q)}\vert}$.

Let's denote by $EH(\theta_{m})$ the following assertion:$\forall \theta\lt \theta_{m},\forall A>0, \exists C_{\theta,A},\displaystyle{\sum_{1\le q\le x^{\theta}}E(x,q)\le C_{\theta,A}\dfrac{x}{(\log x)^{A}}}$.

The still unproven Elliott-Halberstam conjecture is $EH(1)$.

Let's denote by $\mathcal{E}_{\theta}(x)$ the quantity $\displaystyle{\sum_{1\le q\le x^{\theta}}\dfrac{\varphi(q)E(x,q)}{\pi(x)}}$ and by $\theta_{s}$ the quantity $\displaystyle{\sup\{\theta, \lim_{x\to\infty} \mathcal{E}_{\theta}(x)\lt\infty\}}$.

Does $EH(\theta_{s})$ hold?

Thanks in advance.