Strange definition of microcanonical partition function

There is quite a big controversy these days about the correct definition of the entropy in the microcanonical ensemble (the debate between the Gibbs and Boltzmann entropy), which is closely related to the question.

Everyone agrees that the correct definition of the density matrix is given by $$\rho(E)=\frac{\delta(E-H)}{\omega(E)},$$ where $H$ is the Hamiltonian and $$ \omega(E)=Tr\,\delta(E-H).$$

Then the question is the correct definition of the entropy. Boltzmann says $S_B=\ln \omega(E)$, whereas Gibbs argued $S_G=\ln \Omega(E)$ where $$ \Omega(E)=\int_0^E\omega(e) de.$$ In the text quoted by the OP, the partition function corresponds to $\Omega(E)$.

Note that in most cases, in the thermodynamics limit, both entropies gives the same result. The question arises in the case of small systems and special cases with bounded from above spectra. Hilbert et al. (arXiv:1408.5382 and arXiv:1304.2066) argue that only the Gibbs entropy is thermodynamically consistent. I must say that I find their arguments compelling, and that of their opponents, given in at least two comments of their papers, not at all.