The leaf-density phase-transition conjecture for optimal resistor networks

Consider optimal graphs with a prescribed average degree, and let a graph have a positive proportion of leaves when the number of degree-one vertices is a positive proportion of its vertices. Write $o(n)$ for a quantity whose ratio to the number $n$ of vertices tends to zero.

Leaf-density phase-transition conjecture. There is a threshold $\alpha_0$ on the average degree such that below it all optimal graphs have a positive proportion of leaves, whereas above it all optimal graphs have $o(n)$ leaves.

This conjecture predicts a phase transition between star-like and regular-like optimal networks. The source says that the analogous behaviour is also expected for graphs that are sufficiently close to optimal, but that stronger expectation is not part of the stated conjecture.