becomes partially massless in the sense of Deser and Waldron [@pmrefs] and the theory is invariant under the gauge transformation $$\label{infinv}\hskip 1cm
 \delta\tilde{f}_{\mu\nu}=\nabla_\mu\nabla_\nu\zeta-\frac{1}{\ell^2}g_{\mu\nu}\zeta \,,$$ with an infinitesimal scalar gauge parameter $\zeta$. Further features of ’partially massless NMG’ were discussed in [@Bergshoeff:2009aq] and [@Grumiller:2010tj].

However, for NMG it was shown that this gauge invariance only appears at the linearized level [@Blagojevic:2011qc] and does not persist in the non-linear theory. We expect the same to be true here and it would be interesting to verify this. Furthermore, we would like to point out the relation to the “canonical bifurcation” [@Deser:2012ci] effect taking place in the three-dimensional pure quadratic curvature model of [@Deser:2009hb]. In fact, for PET gravity in the above mentioned limit all conditions (i)–(iv) of [@Deser:2012ci] for the “canonical bifurcation” are met.

Another remark concerns the values of the masses of the propagating modes. This is directly related to the background around which we linearize the theory. So far we have considered our background to be AdS. In [@Bergshoeff:2012ev] the limit $b_2\to\infty$ is not contained in the physical range of the parameters $\beta$ and $b_2$ because (one of) the mass squared is negative. This is a consequence of choosing an AdS 