We find a mechanism by which antibranes placed in a warped deformed conifold throat can destroy the stabilization of the size of the sphere at the tip, collapsing it to zero size. This conifold destabilization mechanism can be avoided by turning on a large amount of flux on the sphere, but tadpole cancelation makes this incompatible with a hierarchy of scales in a Type IIB flux compactification. This indicates that antibrane uplift cannot be used to construct stable de Sitter vacua with a small cosmological constant in perturbative String Theory. The values of V and V′ for these KKLT‐like scenarios can be parametrically small, but we find that V′/V is still consistent with the de Sitter swampland conjecture. Our results also suggest that there should exist a Klebanov‐Strassler black hole, holographically dual to a deconfined phase with spontaneously broken chiral symmetry.