  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.
