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Enhanced emission from the dense gas tracer HCN (relative to HCO$^+ $) hasbeen proposed as a signature of active galactic nuclei (AGN). In a previoussingle-dish millimeter line survey we identified galaxies with HCN/HCO$ ^+ $(1-0) intensity ratios consistent with those of many AGN but whose mid-infraredspectral diagnostics are consistent with little to no ( $\lesssim15\% $)contribution of an AGN to the bolometric luminosity. To search for putativeheavily obscured AGN, we present and analyze \nustar hard X-ray (3-79 keV)observations of four such galaxies from the Great Observatories All-sky LIRGSurvey. We find no X-ray evidence for AGN in three of the systems and placestrong upper limits on the energetic contribution of any heavily obscured($N_{\rm H}>10^{24}$ cm$^{-2}$) AGN to their bolometric luminosity. The X-rayflux upper limits are presently an order of magnitude below what XDR-drivenchemistry model predict are necessary to drive HCN enhancements. In a fourthsystem we find a hard X-ray excess consistent with the presence of an AGN, butcontributing only $\sim3\%$ of the bolometric luminosity. It is also unclear ifthe AGN is spatially associated with the HCN enhancement. We further explorethe relationship between HCN/HCO$^+$ (for several $\mathrm{J}_\mathrm{upper}$levels) and $L_\mathrm{AGN}/L_\mathrm{IR}$ for a larger sample of systems inthe literature. We find no evidence for correlations between the line ratiosand the AGN fraction derived from X-rays, indicating that HCN/HCO$^+$ intensityratios are not driven by the energetic dominance of AGN, nor are they reliableindicators of whether SMBH accretion is ongoing.

A Hard X-ray Test of HCN Enhancements as a Tracer of Embedded Black Hole Growth