Uniform lower-bound and convergence conjecture for the Bers constants of metric graphs

For each rank $r\geq 3$, let $C_r$ be the constant from the theorem asserting that every rank-$r$ metric graph contains a proper subgraph whose entropy is at least $C_r$. Conjecture on the constants $C_r$. The constant $C_r$ is universally bounded from below in $r$, and moreover converges to $1$ as $r\to\infty$. The source proves convergence to $1$ for roses and conjectures the corresponding statement for general metric graphs.