  We study the six-point NMHV ratio function in planar ${\cal N}=4$ SYM theory
in the context of positive geometry. The Amplituhedron construction of the
integrand for the amplitudes provides a kinematical region in which the
integrand was observed to be positive. It is natural to conjecture that this
property survives integration, i.e. that the final result for the ratio
function is also positive in this region. Establishing such a result would
imply that preserving positivity is a surprising property of the Minkowski
contour of integration and it might indicate some deeper underlying structure.
We find that the ratio function is positive everywhere we have tested it,
including analytic results for special kinematical regions at one and two
loops, as well as robust numerical evidence through five loops. There is also
evidence for not just positivity, but monotonicity in a "radial" direction. We
also investigate positivity of the MHV six-gluon amplitude. While the remainder
function ceases to be positive at four loops, the BDS-like normalized MHV
amplitude appears to be positive through five loops.
