Let $\pi$ be an automorphic form on $\mathrm{GL}(3,\mathbb{A}_{\mathbb{Q}})$. 
Do we know any case that
\\[\int_0^{T} \left|L(\frac{1}{2} + it, \pi)\right| dt \gg T\\]
holds unconditionally?

I know the conjectured asymptotic formula is $T \log^* T$.