Many thanks for explaining the history of this direction and the development of this topic with new references. But what's astonishing is that the lower bounds for $L(1+it, \pi)$ at $\Re s=1$ and $L(1, \pi)$ at $s=1$ are totally different! Thanks.

Thanks for timely reply. Great thanks. It is really illuminating to communicate with you.

Thanks for clarification of my these blurs. Much obliged. Could you give some more comments on how to understand "...if at the same time the exponent of 𝑞 is enlarged to 1/2+𝜀" in you post? Thanks.