  We prove that every Gauduchon metric on an Inoue-Bombieri surface admits a
strongly leafwise flat form in its $\partial\overline\partial$-class. Using
this result, we deduce uniform convergence of the normalized Chern-Ricci flow
starting at any Gauduchon metric on all Inoue-Bombieri surfaces. We also show
that the convergence is smooth with bounded curvature for initial metrics in
the $\partial\overline\partial$-class of the Tricerri/Vaisman metric.
