  Recently, various examples of asymptotically flat, rotating black holes (BHs)
with synchronized hair have been explicitly constructed, including Kerr BHs
with scalar or Proca hair, and Myers-Perry BHs with scalar hair and a mass gap,
showing there is a general mechanism at work. All these solutions have been
found numerically, integrating the fully non-linear field equations of motion
from the event horizon outwards. Here, we address the spacetime geometry of
these solutions inside the event horizon. Firstly, we provide arguments, within
linear theory, that there is no regular inner horizon for these solutions.
Then, we address this question fully non-linearly, using as a tractable model
five dimensional, equal spinning, Myers-Perry hairy BHs. We find that, for
non-extremal solutions: $(1)$ the inside spacetime geometry in the vicinity of
the event horizon is smooth and the equations of motion can be integrated
inwards; $(2)$ before an inner horizon is reached, the spacetime curvature
grows (apparently) without bound. In all cases, our results suggest the absence
of a smooth Cauchy horizon, beyond which the metric can be extended, for hairy
BHs with synchronized hair.
