Here we show in simplified simulations that the (ir)reversibility of the retreat of instability-prone, Antarctica-type glaciers can strongly depend on the depth of the bed depression they rest on. If it is sufficiently deep, then the destabilized glacier does not recover from its collapsed state. Our results suggest that glaciers resting on a wide and deep bed depression, such as Antarctica's Thwaites Glacier, are particularly susceptible to irreversible retreat.