Mean-value Lindelöf conjecture for real Dirichlet characters

Let $k\geqslant1$ be an integer, and let $Q,T\geqslant2$ be real numbers. Let $\mathcal{O}_2(Q)$ denote the family of real Dirichlet characters counted in the paper, and let $L(s,\chi)$ be the associated Dirichlet $L$-function. For $t\in[-T,T]$, consider the averaged moment

$$\sum_{\chi\in\mathcal{O}_2(Q)}\int_{-T}^{T}\left|L\left(\tfrac12+it,\chi\right)\right|^{2k}\,\mathrm{d}t.$$

Mean-value Lindelöf conjecture. For every $\varepsilon>0$,

$$\sum_{\chi\in\mathcal{O}_2(Q)}\int_{-T}^{T}\left|L\left(\tfrac12+it,\chi\right)\right|^{2k}\,\mathrm{d}t\ll (QT)^{1+\varepsilon},$$

where the implied constant depends on $k$ and $\varepsilon$ at most. This is formulated as an averaged Lindelöf hypothesis for $L$-functions associated with real characters; the paper presents it as a conjectural strengthening of the preceding fixed-order moment bound.