  We study the impact of deviations from the Kennicutt-Schmidt relation
(quantified by the `burstiness' parameter $\kappa_s$), gas metallicity ($Z$),
and density ($n$) on the observed [OIII]88$\mu$m/[CII]158$\mu$m surface
brightness ratios ($\Sigma_{[OIII]}/\Sigma_{[CII]}$) in nine galaxies at
$z\approx6-9$. We first discuss possible biases in the measured
$\Sigma_{[OIII]}/\Sigma_{[CII]}$ ratios by comparing the data with zoom-in
cosmological simulations, and then use a Markov Chain Monte Carlo algorithm to
derive the best fit values of ($\kappa_s, Z, n$). We find that (i) the
strongest dependence of $\Sigma_{[OIII]}/\Sigma_{[CII]}$ is on $\kappa_s$; (ii)
high ratios identify starburst galaxies with short gas depletion times
($t_{dep}=6-49\,\rm Myr$); (iii) a secondary dependence on density is found,
with $\Sigma_{[OIII]}/\Sigma_{[CII]}$ anticorrelating with $n$ as a result of
the lower [OIII] critical density, (iv) the ratio only weakly depends on $Z$.
The nine galaxies are significantly enriched (Z=0.2-0.5 $Z_\odot$), and dense
($n=10^{1-3} {\rm cm}^{-3}$). This lends further support to the starburst
scenario in which a rapid enrichment of the interstellar medium is expected.
