Shin–Templier's average conductor growth conjecture for functorial L-functions

Let $\rho$ be an $L$-homomorphism and let $C_{\rho,X}$ denote the average analytic conductor associated with the family indexed by representations with analytic conductor bounded by $X$. Average conductor growth conjecture. There exist nonnegative constants $\mathfrak{c}(\rho)$ and $\mathfrak{C}(\rho)$ such that, for all sufficiently large $X>1$,

$$X^{\mathfrak{c}(\rho)}\ll C_{\rho,X}\ll X^{\mathfrak{C}(\rho)}.$$

This is the polynomial-growth hypothesis needed to control the low-lying zero statistic; the source does not establish it for all $L$-homomorphisms.