Slowly sheared particulate media like sand and suspensions flow heterogeneously as they yield via shear bands, in which most strain accumulates. Understanding shear band localization from microscopics is still a major challenge. One class of so-called non-local theories identified that the width of the shearing zone $\xi$ should depend on the stress field, in particular through the local distance to the yield point of the material. We explicitly test this stress sensitivity picture by using a uniquely stress-tunable suspension while probing its flow behavior in a classic geometry in which shear bands are known to scale nontrivially with local stress: the Split-Bottom Shear Cell (SBSC). The stress-tunable suspension is composed of mildly polydisperse soft, slippery hydrogel spheres submersed in water. We measure their flow profiles and rheology while controlling the confinement stress via both hydrostatic effects and compression. Unique for these soft particles is that we can probe flow fields under confining normal stresses that reach about 1% of their elastic modulus. We determine the average angular velocity profiles in the quasi-static flow regime using Magnetic Resonance Imaging based particle image velocimetry (MRI-PIV) and discrete element method (DEM) simulations. We explicitly match a pressure-sensitive non-local granular fluidity (NGF) model to observed flow behavior. We find that shear bands for this type of suspension become extremely broad under the low confining stresses from the almost density-matched fluid particle mixture, while collapsing to a narrow shear zone under finite, externally imposed compression levels. The DEM and NGF results match the observations qualitatively, confirming the conjectured pressure sensitivity for suspensions and its role in the NGF model. Our results indicate that pressure sensitivity should be part of non-local flow rules to describe slow flows of granular media.