Equality of minimum and maximum layering cluster-diameters

Determine whether the minimum and maximum cluster-diameters over all layering partitions of every graph are equal, that is, whether \Delta(G)=\widehat{\Delta}(G).

Background

For a graph G, \Delta(G) is the minimum cluster-diameter over all start vertices, whereas \widehat{\Delta}(G) is the maximum. The paper proves that these parameters differ by at most a factor of three and asks whether the constants in its various coarse-equivalence inequalities can be improved, including whether these two parameters always coincide.