Hypersurface Orthogonality of Conformal Killing field on Conformal Killing Horizon

Suppose we have a spacetime manifold $(mathcal{M},g)$ admitting a conformal Killing Horizon $mathcal{H}_c$ generated by a conformal Killing field $chi^a$ (which is null only on the conformal Killing horizon). The conformal killing field is defined by the equation,
begin{equation}
mathcal{L}_{chi}g_{ab} = 2 psi g_{ab} ~,
end{equation}
where $psi$ is a scalar field. The norm of $chi$ is zero exactly only on $mathcal{H}_c$ i.e $chi^a chi_a overset{mathcal{H}_c} = 0$. The conformal Killing field is a null geodesic generator of the conformal killing horizon and hence satisfies,
begin{equation}
chi^a nabla_a chi_b overset{mathcal{H}_c} = kappa chi_b ~,
end{equation}
where $kappa$ is the non-affinity parameter of the null geodesic generator $chi^a$. My question is whether $chi^a$ is hypersurface orthogonal to the conformal Killing horizon, i.e whether we have the validity of the statement that,
begin{equation}
chi_{[a} nabla_b chi_{c]} overset{mathcal{H}_c} = 0 ~?
end{equation}
The last result would have been been true had $chi^a$ been a Killing vector field generating a Killing Horizon. Any help or references towards this would be very helpful.