  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.
