  The possibility that an unconventional depletion in the center of the charge
density distribution of certain nuclei occurs due to a purely quantum
mechanical effect has attracted theoretical and experimental attention in
recent years. We report on ab initio self-consistent Green's function
calculations of one of such candidates, $^{34}$Si, together with its Z+2
neighbour $^{36}$S. Binding energies, rms radii and density distributions of
the two nuclei as well as low-lying spectroscopy of $^{35}$Si, $^{37}$S,
$^{33}$Al and $^{35}$P are discussed. The interpretation of one-nucleon removal
and addition spectra in terms of the evolution of the underlying shell
structure is also provided. The study is repeated using several chiral
effective field theory Hamiltonians as a way to test the robustness of the
results with respect to input inter-nucleon interactions. The prediction
regarding the (non-)existence of the bubble structure in $^{34}$Si varies
significantly with the nuclear Hamiltonian used. However, demanding that the
experimental charge density distribution and the root mean square radius of
$^{36}$S are well reproduced, along with $^{34}$Si and $^{36}$S binding
energies, only leaves the NNLO$_{\text{sat}}$ Hamiltonian as a serious
candidate to perform this prediction. In this context, a bubble structure,
whose fingerprint should be visible in an electron scattering experiment of
$^{34}$Si, is predicted. Furthermore, a clear correlation is established
between the occurrence of the bubble structure and the weakening of the
1/2$^-$-3/2$^-$ splitting in the spectrum of $^{35}$Si as compared to $^{37}$S.
