The shape distortion of the presumed Milky Way dark matter halo can impact the local density of dark matter and thus the direct detection program.
We examine the population of galactic rotation curves measured by SPARC and fit them
to dark matter haloes that are distorted with a multipole density distribution, finding a significantly better fit with prolate haloes over spherically symmetric ones.
This is to be expected since the long-distance Rubin flattening $v(r)= {\rm constant}$ is the natural Kepler law due to a filamentary rather than a spherical source. Then, elongating the distribution brings about a smaller $\chi^2$, all other things being equal, including the
use of several different radial dark matter profiles.
The ellipticities that we fit to rotation curve data seem to be much more significant than those
computed in cosmological simulations of dark matter haloes.
If the Milky Way's halo would be typical of the spiral-galaxy SPARC sample (which we presently ignore), the local dark matter density might currently be overestimated by a factor 2. This would carry on into dark-matter cross-section bounds.