  Observational searches for faint active nuclei at $z > 6$ have been extremely
elusive, with a few candidates whose high-$z$ nature is still to be confirmed.
Interpreting this lack of detections is crucial to improve our understanding of
high-$z$ supermassive black holes (SMBHs) formation and growth. In this work,
we present a model for the emission of accreting BHs in the X-ray band, taking
into account super-Eddington accretion, which can be very common in gas-rich
systems at high-$z$. We compute the spectral energy distribution for a sample
of active galaxies simulated in a cosmological context, which represent the
progenitors of a $z \sim 6$ SMBH with $M_{\rm BH} \sim 10^9 \, M_\odot$. We
find an average Compton thick fraction of $\sim 45\%$ and large typical column
densities ($N_H \gtrsim 10^{23} \rm \, cm^2$). However, faint progenitors are
still luminous enough to be detected in the X-ray band of current surveys. Even
accounting for a maximum obscuration effect, the number of detectable BHs is
reduced at most by a factor 2. In our simulated sample, observations of faint
quasars are mainly limited by their very low active fraction ($f_{\rm act} \sim
1 \%$), which is the result of short, super-critical growth episodes. We
suggest that to detect high-$z$ SMBHs progenitors, large area surveys with
shallower sensitivities, such as Cosmos Legacy and XMM-LSS+XXL, are to be
preferred with respect to deep surveys probing smaller fields, such as CDF-S.
