  Protoclusters at $z>2$ are gas-rich regions characterized by high
star-formation activity. The same physical properties that enhance star
formation in protoclusters are also thought to boost the growth of SMBHs. We
aim to test this scenario by probing the AGN content of SPT2349-56, a massive,
gas-rich, and highly star-forming protocluster core at $z=4.3$ discovered as an
overdensity of DSFGs, via Chandra (200 ks) observations, and comparing the
results with the field environment. We detected two protocluster members,
corresponding to an AGN fraction among DSFGs of $\approx10\%$. This value is
consistent with other protoclusters at $z=2-4$, but higher than the AGN
incidence among DSFGs in the field environment. Both AGN are heavily obscured
sources and hosted in star-forming galaxies with
$\approx3\times10^{10}\,\mathrm{M_\odot}$ stellar masses. We estimate that the
ISM in the host galaxies can contribute significantly to the nuclear
obscuration. One of the two AGN is highly luminous
($L_X=2\times10^{45}\,\mathrm{erg\,s^{-1}}$) and Compton-thick
($N_H=2\times10^{24}\,\mathrm{cm^{-2}}$), and likely powered by a
$M_{BH}>6\times10^8\,\mathrm{M_\odot}$ SMBH. Its high accretion rate suggests
that it is in the phase of efficient growth required to explain the presence of
extremely massive SMBHs in the centers of local galaxy clusters. Considering
SPT2349-56 and DRC, a similar protocuster at $z=4$, we find that gas-rich
protocluster cores at $z\approx4$ enhance the triggering of luminous
(log$\frac{L_X}{\mathrm{erg\,s^{-1}}}=45-46$) AGN by 3-5 orders of magnitude
with respect to the field environment. Our results indicate that gas-rich
protoclusters at high redshift boost the growth of SMBHs, which will likely
impact the subsequent evolution of the structures, and thus represent key
science targets to obtain a complete understanding of the relation between
environment and galaxy evolution.
