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