Nanoscale confinement has been shown to alter the glass transition and associated mechanical and transport properties of glass-forming materials. Inspired by expected interrelations between nanoconfinement effects, cooperative dynamics in supercooled liquids, and the "fragility" (or temperature-abruptness) of the glass transition, it is commonly expected that nanoconfinement effects on Tg should be more pronounced for more fragile glass formers. Here we employ molecular dynamics simulations of glass formation in the bulk and under nanoconfinement of model polymers in which we systematically tune fragility by several routes. Results indicate that a correlation between fragility and the strength of nanoconfinement effects is weak to modest at best when considering all systems but can appear to be stronger when considering a subset of systems. This outcome is consistent with a reanalysis of the Adam-Gibbs theory of glass formation indicating that fragility does not necessarily track in a universal way with the scale of cooperative motion in glass-forming liquids. Finally, we find that factors such as composition gradients or variability in measurement sensitivity to different parts of the dynamic gradient have the potential to significantly confound efforts to identify trends in Tg-nanoconfinement effects with variables such as fragility, emphasizing the importance of employing diverse data sets and multiple metrologies in the study of this problem.