will comment on the implications of such a restriction.

Hand-picked partition functions of critical gravities
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If we do not take into account the contribution from the spin-two modes the partition function of critical NMG would become $$\label{toomuch}\hskip 1cm
 Z_{\text{``no modes''}}=\prod_{m\geq3}^\infty\prod_{n\geq0}^\infty |1-q^{m+n}|^2\,.$$ Hence, we would be lead to conclude that the theory deprived of its spin-two modes is not trivial. However, it seems likely that the correct interpretation is that here we truncated too much and the resulting theory does not make sense.[^5]

If we were to cancel only the log-modes we find, in fact, for any parity even critical gravity theory without log-modes, $$\hskip 1cm
 \tilde{Z}_{\rm cNMG}=\sqrt{\frac{\det(-\nabla^2+2)_1^T}{\det(-\nabla^2-2)_2^{TT}}}=Z_{\rm Ein}\,.$$ Thus, we could conclude that critical NMG with Brown–Henneaux boundary conditions is dual to Einstein gravity. However, we know that the only propagating modes are not only pure gauge but also null states and the dual CFT has vanishing central charge. Therefore, according to [@Castro:2011ui] we would have to drop also the Einstein modes which, naively, brings us back to (\[toomuch\]).

Let us now consider a six-derivative theory that offers “higher-rank” criticality and more possibilities to truncate the theory, PET gravity. Truncating to the zero charge sub-sector, i.e. imposing log boundary 