  More than $200$ quasars have been detected so far at $z > 6$, with only one
showing clear signs of strong gravitational lensing. Some studies call for a
missing population of lensed high-$z$ quasars, but their existence is still in
doubt. A large fraction of high-$z$ quasars being lensed would have a
significant effect on the shape of the intrinsic quasar luminosity function
(QLF). Here, we perform the first systematic search for lensed X-ray-detected
quasars at $z \gtrsim 6$ employing a Bayesian analysis, with the code BAYMAX,
to look for morphological evidence of multiple images that may escape a visual
inspection. We analyzed a sample of 22 quasars at $z > 5.8$ imaged by the
Chandra X-ray observatory and found none with statistically significant
multiple images. In the sub-sample of the 8 sources with photon counts $>20$ we
exclude multiple images with separations $r>1''$ and count ratios $f>0.4$, or
with separations as small as $0.''7$ and $f>0.7$ at $95\%$ confidence level.
Comparing this non-detection with predictions from theoretical models
suggesting a high and a low lensed fraction, we placed upper limits on the
bright-end slope, $\beta$, of the QLF. Using only the sub-sample with 8
sources, we obtain, in the high-lensing model, a limit $\beta < 3.38$. Assuming
no multiple source is present in the full sample of 22 sources, we obtain
$\beta < 2.89$ and $\beta < 3.53$ in the high and low lensing models,
respectively. These constraints strongly disfavor steep QLF shapes previously
proposed in the literature.
