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Context.Large-scale environment is one of the main physical drivers of galaxy evolution. The densest regions at high redshifts (i.e.z > 2 protoclusters) are gas-rich regions characterised by high star formation activity. The same physical properties that enhance star formation in protoclusters are also thought to boost the growth of supermassive black holes (SMBHs), most likely in heavily obscured conditions. Aims.We aim to test this scenario by probing the active galactic nucleus (AGN) content of SPT2349–56: a massive, gas-rich, and highly star-forming protocluster core atz = 4.3 discovered as an overdensity of dusty star-forming galaxies (DSFGs). We compare our results with data on the field environment and other protoclusters. Methods.We observed SPT2349–56 withChandra(200 ks) and searched for X-ray emission from the known galaxy members. We also performed a spectral energy distribution fitting procedure to derive the physical properties of the discovered AGNs. Results.In the X-ray band, we detected two protocluster members: C1 and C6, corresponding to an AGN fraction among DSFGs in the structure of ≈10%. This value is consistent with other protoclusters atz  =  2 − 4, but higher than the AGN incidence among DSFGs in the field environment. Both AGNs are heavily obscured sources, hosted in star-forming galaxies with ≈3 × 10¹⁰ M_⊙stellar masses. We estimate that the intergalactic medium in the host galaxies contributes to a significant fraction (or even entirely) to the nuclear obscuration. In particular, C1 is a highly luminous (L_X = 2 × 10⁴⁵ erg s⁻¹) and Compton-thick (N_H = 2 × 10²⁴ cm⁻²) AGN, likely powered by aM_BH > 6 × 10⁸ M_⊙SMBH, assuming Eddington-limited accretion. Its high accretion rate suggests that it is in the phase of efficient growth that is generally required to explain the presence of extremely massive SMBHs in the centres of local galaxy clusters. Considering SPT2349–56 and DRC, a similar protocuster atz = 4, and under different assumptions on their volumes, we find that gas-rich protocluster cores atz ≈ 4 enhance the triggering of luminous (logL_X/erg s⁻¹ = 45 − 46) AGNs by three to five orders of magnitude with respect to the predictions from the AGN X-ray luminosity function at a similar redshift in the field environment. We note that this result is not solely driven by the overdensity of the galaxy population in the structures. Conclusions.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. Therefore, they stand as key science targets to obtain a complete understanding of the relation between the environment and galaxy evolution. Dedicated investigations of similar protoclusters are required to definitively confirm this conclusion with a higher statistical significance.