Over the past thirty years, two bodies of literature have developed in parallel presenting mutually exclusive views of the Sun's upper transition region. One model holds that the Sun's upper-transition-region plasmas are confined primarily in hydrostatic funnels with a substantial backheating component. The other model holds that discrete structures, which are effectively isolated from the corona, predominate in the Sun's upper transition region. Purveyors of the latter position have recently begun to present near-resolved observations of discrete structures. The funnel scenario, in contrast, has only been addressed by modeling unresolved upper transition region emission. To address this paradox we have constructed hydrostatic funnel models and tested them against a wider set of solar observations than previously performed. We reproduce the results of the previous analyses, yet find that the hydrostatic funnels are unable to self-consistently match the wider set of observations against which we test the models. We show that it is not possible for a class of funnels having peak temperatures in the transition region or in the corona to match the observations. We conclude that it is implausible that a class of hydrostatic funnels constitutes the dominant emitting component of the Sun's upper-transition-region plasmas as has been suggested.