Finiteness and convergence conjecture for tree network flow

Consider the evolution of a tree under network flow. A singular time is a time at which the flow develops a singularity, and a boundary curve is a curve of the network ending at the boundary. A minimal network is a network satisfying the relevant minimality condition, allowing degeneration. Finiteness and convergence conjecture. The number of singular times during the evolution of a tree is finite. If no boundary curve disappears during the evolution, then the flow exists for all positive times and converges to a possibly degenerate minimal network. This is presented as a reasonable expectation for general tree-like networks, with appropriate modifications, rather than as a result established by the paper's special symmetric case.