Local Lindelöf hypothesis for Whittaker newforms

Let $\pi$ be the representation under consideration, with central character $\omega_\pi$ of conductor exponent $a(\omega_\pi)$, and let $n_1$ and $m_1$ be the parameters used in the paper, so that $a(\omega_\pi)\leq n_1$ is equivalent to $m_1=0$. Let $G$ be the relevant group and $W_\pi$ the Whittaker newform. Local Lindelöf hypothesis for Whittaker newforms. If

$$a(\omega_\pi)\leq n_1 \quad\text{(equivalently, }m_1=0\text{)},$$

then

$$1\ll \sup_{g\in G}\lvert W_\pi(g)\rvert\ll_\varepsilon q^{n\varepsilon}.$$

This conjecture predicts a near-uniform upper bound for the sup norm of the Whittaker newform when the central character has sufficiently small conductor; the supplied excerpt does not indicate whether it has been resolved.