The maximum critical-value conjecture for families of L-functions

Let $\mathcal F$ be a family of $L$-functions, and for $F\in\mathcal F$ let $c(F)$ denote its conductor. Assume the family is ordered by conductor and has approximately $D$ members with conductor below $D$. For unitary families set $B=1/2$, and for symplectic or orthogonal families set $B=1$. Maximum critical-value conjecture.

$$\max_{\substack{F\in\mathcal F\c(F)\le D}}|F(\tfrac12)|=\exp\left((1+o(1))\sqrt{B\log D\log\log D}\right).$$

The implied constant depends only on $\mathcal F$. This extends the zeta maximum-size prediction to families of $L$-functions, with the symmetry type determining the constant in the exponent.