A theoretical investigation of far-infrared fine structure lines at $z>6$ and of the origin of the [OIII]88/[CII]158 enhancement

[Abridged] The [OIII]${}_{88\mu m}$/[CII]${}_{158\mu m}$ and [OIII]${}_{88\mu m}$/[NII]${}_{122\mu m}$ luminosity ratios have shown to be promising tracers of the ionisation state and gas-phase metallicity of the ISM. Observations of galaxies at redshift $z>6$ show peculiarly higher [OIII]${}_{88\mu m}$/[CII]${}_{158\mu m}$ luminosity ratios compared to local sources. No model has so far successfully managed to match the observed emission from both [OIII]${}_{88\mu m}$ and [CII]${}_{158\mu m}$ as well as their ratio. We use Cloudy to model the [CII]${}_{158\mu m}$, [OIII]${}_{88\mu m}$, [NII]${}_{122\mu m}$ and [NIII]${}_{57\mu m}$ emission lines of Ponos: a high-resolution ($m_{\mathrm{g}\mathrm{a}\mathrm{s}}=883.4M_{\odot }$) cosmological zoom-in simulation of a galaxy at redshift $z=6.5$, which is post-processed using kramses-rt. We modify Carbon, Nitrogen and Oxygen abundances in our Cloudy models to obtain C/O and N/O abundance ratios respectively lower and higher than Solar, more in line with recent high-z observational constraints. We find [OIII]${}_{88\mu m}$/[CII]${}_{158\mu m}$ luminosity ratios that are a factor of $\sim 5$ higher compared to models assuming solar abundances. Additionally, we find an overall better agreement of the simulation with high-z observational constraints of the [CII]${}_{158\mu m}$-SFR and [OIII]${}_{88\mu m}$-SFR relations. This shows that a lower C/O abundance ratio is essential to reproduce the enhanced [OIII]${}_{88\mu m}$/[CII]${}_{158\mu m}$ luminosity ratios observed at $z>6$. By assuming a super-solar N/O ratio, motivated by recent $z>6$ JWST observations, our models yield an [OIII]${}_{88\mu m}$/[NII]${}_{122\mu m}$ ratio of $1.3$, which, according to current theoretical models, would be more appropriate for a galaxy with a lower ionisation parameter than the one we estimated for Ponos.