The global sup-norm conjecture for newforms with mildly ramified central characters

Let $F$ be a number field, let $G=\operatorname{GL}_2$, and let $\pi$ be an automorphic representation with finite conductor $N$ and local conductor exponents $n_v$ and central-character conductor exponents $m_v$. Let $\phi\in\pi$ be a newform normalized by $\|\phi\|_2=1$. The global sup-norm conjecture. If $m_v\leq\lceil n_v/2\rceil$ for every place $v$, then

$$N^{-\varepsilon}\ll_{\pi_\infty,F,\varepsilon}\sup_{g\in G(\mathbb{A})}|\phi(g)|\ll_{\pi_\infty,F,\varepsilon}N^{\varepsilon}.$$

This optimistic conjecture combines the local Whittaker hypothesis with the expectation that local quantities provide the only obstructions to an essentially minimal global sup norm. The paper does not claim a proof of either bound in this generality.