  Despite significant progress both observationally and theoretically, the
origin of high-ionization nebular HeII emission in galaxies dominated by
stellar photoionization remains unclear. Accretion-powered radiation from
high-mass X-ray binaries (HMXBs) is still one of the leading proposed
explanations for the missing $\mathrm{He^+}$-ionizing photons, but this
scenario has yet to be conclusively tested. In this paper, we present nebular
line predictions from a grid of photoionization models with input SEDs
containing the joint contribution of both stellar atmospheres and a multi-color
disk model for HMXBs. This grid demonstrates that HMXBs are inefficient
producers of the photons necessary to power HeII, and can only boost this line
substantially in galaxies with HMXB populations large enough to power X-ray
luminosities of $10^{42}$ erg/s per unit star formation rate (SFR). To test
this, we assemble a sample of eleven low-redshift star-forming galaxies with
high-quality constraints on both X-ray emission from Chandra and HeII emission
from deep optical spectra, including new observations with the MMT. These data
reveal that the HMXB populations of these nearby systems are insufficient to
account for the observed HeII strengths, with typical X-ray luminosities or
upper limits thereon of only $10^{40}$-$10^{41}$ erg/s per SFR. This indicates
that HMXBs are not the dominant source of $\mathrm{He^+}$ ionization in these
metal-poor star-forming galaxies. We suggest that the solution may instead
reside in revisions to stellar wind predictions, softer X-ray sources, or very
hot products of binary evolution at low metallicity.
