  We present an analysis searching for dual AGN among 62 high-redshift ($2.5 <
z < 3.5$) X-ray sources selected from publicly available deep Chandra fields.
We aim to quantify the frequency of dual AGN in the high-redshift Universe,
which holds implications for black hole merger timescales and low-frequency
gravitational wave detection rates. We analyze each X-ray source using BAYMAX,
an analysis tool that calculates the Bayes factor for whether a given archival
Chandra AGN is more likely a single or dual point source. We find no strong
evidence for dual AGN in any individual source in our sample. We then increase
our sensitivity to search for dual AGN across the sample by comparing our
measured distribution of Bayes factors to that expected from a sample composed
entirely of single point sources, and again find no evidence for dual AGN in
the observed sample distribution. Although our analysis utilizes one of the
largest Chandra catalogs of high-$z$ X-ray point sources available to study,
the findings remain limited by the modest number of sources observed at the
highest spatial resolution with Chandra and the typical count rates of the
detected sources. Our non-detection allows us to place an upper-limit on the
X-ray dual AGN fraction between $2.5<z<3.5$ of 4.8\%. Expanding substantially
on these results at X-ray wavelengths will require future surveys spanning
larger sky areas and extending to fainter fluxes than has been possible with
Chandra. We illustrate the potential of the AXIS mission concept in this
regard.
