Does the Guth--Maynard zero-density estimate imply a $T^{5/9+\varepsilon}$ bound for a logarithmic integral of $\zeta(s)$?

Fix $\frac12<\sigma<1$, and define the signed logarithmic integral

$$ A_\sigma(T) \int_2^T \log |\zeta(\sigma+it)|,dt. $$

I am interested in transferring recent zero-density estimates into bounds for $A_\sigma(T)$.

Applying Littlewood's lemma to $\zeta(s)$ in the rectangle

$$\sigma\le \operatorname{Re}s\le c, \qquad 2\le \operatorname{Im}s\le T, $$

where $c>1$, appears to give

$$ A_\sigma(T) 2\pi \sum_{\substack{2<\gamma\le T\ \beta>\sigma}} (\beta-\sigma) + O_\sigma(\log T), $$

after controlling the integral on $\operatorname{Re}s=c$ and the two horizontal boundary terms. Here $\rho=\beta+i\gamma$ runs over the nontrivial zeros of $\zeta(s)$, counted with multiplicity.

Since every nontrivial zero satisfies $\beta<1$, one has

$$ \sum_{\substack{2<\gamma\le T\ \beta>\sigma}} (\beta-\sigma) \le (1-\sigma)N(\sigma,T). $$

Guth and Maynard recently proved

$$ N(\sigma,T) \le T^{\frac{15(1-\sigma)}{3+5\sigma}+o(1)}. $$

Consequently, it seems that

$$ A_\sigma(T) \ll_{\sigma,\varepsilon} T^{\frac{15(1-\sigma)}{3+5\sigma}+\varepsilon}. $$

In particular, at $\sigma=\frac34$,

$$ \frac{15(1-\sigma)}{3+5\sigma} \frac59, $$

so this would give

$$ \boxed{ \int_2^T \log\left| \zeta\left(\frac34+it\right) \right|,dt \ll_\varepsilon T^{5/9+\varepsilon}. } $$

The corresponding transfer from the classical Ingham--Huxley zero-density estimates gives the exponent $3/5$ at $\sigma=3/4$. Thus the new estimate appears to improve the exponent by

$$ \frac35-\frac59=\frac{2}{45}. $$

More generally, the Guth--Maynard exponent improves the classical zero-density envelope in the range

$$ \frac{7}{10}<\sigma<\frac45, $$

and therefore appears to give an improved bound for $A_\sigma(T)$ throughout this interval.

My questions are:

1. Is this application of Littlewood's lemma correct, including the claimed $O_\sigma(\log T)$ contribution from the remaining boundary terms?

2. Does the Guth--Maynard estimate therefore imply

$$ A_\sigma(T) \ll_{\sigma,\varepsilon} T^{\frac{15(1-\sigma)}{3+5\sigma}+\varepsilon}? $$

3. Has this consequence already appeared in the literature? If hasn't, does this result have a merit as a short note ?

Reference: L. Guth and J. Maynard, New large value estimates for Dirichlet polynomials, Theorem 1.2.