The discovery of hundreds of QSOs in the first Gyr of the Universe powered by already grown SMBHs challenges our knowledge of SMBH formation. In particular, investigations of $z>6$ QSOs presenting notable properties can provide unique information on the physics of fast SMBH growth in the early universe. We present the results of follow-up observations of the $z=6.515$ radio-quiet QSO PSO167-13, which is interacting with a close companion galaxy. The PSO167-13 system has been recently proposed to host the first heavily obscured X-ray source at high redshift. We observed PSO167-13 with Chandra/ACIS-S (177 ks), and obtained new spectroscopic observations (7.2 h) with Magellan/FIRE. No significant X-ray emission is detected from the PSO167-13 system, suggesting that the obscured X-ray source previously tentatively detected was either due to a strong background fluctuation or is highly variable. The upper limit (90% confidence level) on the X-ray emission of PSO167-13 ($L_{2-10,mathrm{keV}}<8.3times10^{43},mathrm{erg s^{-1}}$) is the lowest available for a $z>6$ QSO. The ratio between the X-ray and UV luminosity of $alpha_{ox}<-1.95$ makes PSO167-13 a strong outlier from the $alpha_{ox}-L_{UV}$ and $L_X-L_{mathrm{bol}}$ relations. In particular, its X-ray emission is $>6$ times weaker than the expectation based on its UV luminosity. The new Magellan/FIRE spectrum of PSO167-13 is strongly affected by the unfavorable sky conditions, but the tentatively detected C IV and Mg II emission lines appear strongly blueshifted. The most plausible explanations for the X-ray weakness of PSO167-13 are intrinsic weakness or small-scale absorption by Compton-thick material. The possible strong blueshift of its emission lines hints at the presence of nuclear winds, which could be related to its X-ray weakness.
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