We present Chandra ACIS-S and ATCA radio continuum observations of the strongly lensed dusty, star-forming galaxy SPT-S J034640-5204.9 (hereafter SPT0346-52) at $z$ = 5.656. This galaxy has also been observed with ALMA, HST, Spitzer, Herschel, APEX, and the VLT. Previous observations indicate that if the infrared (IR) emission is driven by star formation, then the inferred lensing-corrected star formation rate ($sim$ 4500 $M_{sun}$ yr$^{-1}$) and star formation rate surface density $Sigma_{rm SFR}$ ($sim$ 2000 $M_{sun} {yr^{-1}} {kpc^{-2}}$) are both exceptionally high. It remained unclear from the previous data, however, whether a central active galactic nucleus (AGN) contributes appreciably to the IR luminosity. The {it Chandra} upper limit shows that SPT0346-52 is consistent with being star-formation dominated in the X-ray, and any AGN contribution to the IR emission is negligible. The ATCA radio continuum upper limits are also consistent with the FIR-to-radio correlation for star-forming galaxies with no indication of an additional AGN contribution. The observed prodigious intrinsic IR luminosity of (3.6 $pm$ 0.3) $times$ 10$^{13}$ $L_{sun}$ originates almost solely from vigorous star formation activity. With an intrinsic source size of 0.61 $pm$ 0.03 kpc, SPT0346-52 is confirmed to have one of the highest $Sigma_{SFR}$ of any known galaxy. This high $Sigma_{SFR}$, which approaches the Eddington limit for a radiation pressure supported starburst, may be explained by a combination of very high star formation efficiency and gas fraction.
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