  Baker and Rumely's tau lower bound conjecture claims that if the tau constant
of a metrized graph is divided by its total length, this ratio must be bounded
below by a positive constant for all metrized graphs. We construct several
families of metrized graphs having small tau constants. In addition to
numerical computations, we prove that the tau constants of the metrized graphs
in one of these families, the hexagonal nets around a torus, asymptotically
approach to $\frac{1}{108}$ which is our conjectural lower bound.
