Twisted fourth-moment hypothesis for Dirichlet $L$-functions

Let $\boldsymbol{\alpha}=(\alpha_m)$ be a coefficient sequence, let $q$ be a positive integer, let $t\in\mathbf{R}$, and let $\delta\in[0,1)$. The hypothesis concerns the twisted fourth moment

$$\frac{1}{\varphi(q)}\sum_{\substack{\chi\, (\operatorname{mod}{q})\\ \chi\neq\chi_0}}\left|\sum_{m\leqslant q^\delta}\alpha_m\chi(m)\right|^2\left|L\left(\tfrac12+it,\chi\right)\right|^4 \ll (q(1+|t|))^{\varepsilon}q^\delta\|\boldsymbol{\alpha}\|_\infty.$$

Twisted fourth-moment hypothesis. For every coefficient sequence $\boldsymbol{\alpha}$ and every $\varepsilon>0$, the displayed estimate holds. This conjecture asks for the averaged Lindelöf-type bound with arbitrarily large twisting length exponent $\delta<1$, and would provide a weaker substitute for the Lindelöf hypothesis in the paper's conditional results. Its status is not resolved in the supplied text.