  In this work we continue the study of the one-loop partition function for
higher derivative conformal higher spin (CHS) fields in six dimensions and its
holographic counterpart given by massless higher spin Fronsdal fields in seven
dimensions. In going beyond the conformal class of the boundary round 6-sphere,
we start by considering a Ricci-flat, but not conformally flat, boundary and
the corresponding Poincar\'e-Einstein spacefilling metric. Here we are able to
match the UV logarithmic divergence of the boundary with the IR logarithmic
divergence of the bulk, very much like in the known 4D/5D setting, under the
assumptions of factorization of the higher derivative CHS kinetic operator and
WKB-exactness of the heat kernel of the dual bulk field. A key technical
ingredient in this construction is the determination of the fourth heat kernel
coefficient b6 for Lichnerowicz Laplacians on both 6D and 7D Einstein
manifolds. These results allow to obtain, in addition to the already known
type-A Weyl anomaly, two of the three independent type-B anomaly coefficients
in terms of the third, say c_3 for instance. In order to gain access to c_3,
and thus determine the four central charges independently, we further consider
a generic non Ricci-flat Einstein boundary. However, in this case we find a
mismatch between boundary and bulk computations for spins higher than two. We
close by discussing the nature of this discrepancy and perspectives for a
possible amendment.
