  Classical single centered solutions of $1/4$ BPS dyons in ${\cal N}=4$
theories are usually constructed in duality frames which contain non-trivial
hair degrees of freedom localized outside the horizon. These modes are in
addition to the fermionic zero modes associated with broken supersymmetry.
Identifying and removing the hair from the $1/4$ BPS index allows us to isolate
the degrees of freedom associated with the horizon. The spherical symmetry of
the horizon then ensures that index of the horizon states has to be positive.
We verify that this is indeed the case for the canonical example of dyons in
type IIB theory on $K3\times T^2$ and prove this property holds for a class of
states. We generalise this observation to all CHL orbifolds, this involves
identifying the hair and isolating the horizon degrees of freedom. We then
identify the horizon states for $1/4$ BPS dyons in ${\cal N}=4$ models obtained
by freely acting ${\mathbb{Z}}_2$ and ${\mathbb{ Z}}_3$ orbifolds of type IIB
theory compactified on $T^6$ and observe that the index is again positive for
single centred black holes. This observation coupled with the fact the $1/4$
BPS index of single centred solutions without removal of the hair violates
positivity indicates that there exists no duality frame in these models without
non-trivial hair.
