Uniform entropy lower bound conjecture for proper subgraphs of metric graphs

Let $r\geq 3$. A metric graph of rank $r$ has unit entropy when its entropy is normalized to equal $1$, and a subgraph is proper when it is a strict subgraph of the graph. Uniform entropy lower bound conjecture. There exists a constant $C>0$ such that every metric graph of rank $r$ with unit entropy contains a proper subgraph with entropy at least $C$. This conjecture would provide a uniform way to move from arbitrary points in the entropy metric space toward completion points represented by unit-entropy metrics on proper subgraphs, supporting an inductive approach to the boundedness question for $(\mathcal{X}^1(\mathbb{F}_r),d_\mathfrak{h})$.