  We investigate in detail recent suggestions that codimension-two braneworlds
in six dimensional supergravity might circumvent Weinberg's no-go theorem for
self-tuning of the cosmological constant. The branes are given finite thickness
in order to regularize mild singularities in their vicinity, and we allow them
to have an arbitrary equation of state. We study perturbatively the time
evolution of the solutions by solving the equations of motion linearized around
a static background. Even allowing for the most general possibility of warping
and nonconical singularities, the geometry does not relax to a static solution
when the brane stress-energies are perturbed. Rather, both the internal and
external geometries become time-dependent, and the system does not exhibit any
self-tuning behavior.
