Logarithmic error bound conjecture for Dirichlet $L$-function zero counts

Let $\chi$ be a character with conductor $q>1$, and define

$$\ell=\log\frac{q(T+2)}{2\pi}.$$

Assume $T\ge5/7$. Logarithmic error bound conjecture. The zero-counting function satisfies

$$\left|N(T,\chi)-\left(\frac{T}{\pi}\log\frac{qT}{2\pi e}-\frac{\chi(-1)}{4}\right)\right|\le\frac{\ell}{\log(2+\ell)}.$$

This is a speculative explicit estimate motivated by Selberg's GRH bound and numerical computation. The supplied text gives no resolution, so the conjecture is open.