  We apply novel, recently developed plasma ray-tracing techniques to model the
propagation of radio photons produced by axion dark matter in neutron star
magnetospheres and combine this with both archival and new data for the
galactic centre magnetar PSR J1745-2900. The emission direction to the observer
and the magnetic orientation are not constrained for this object leading to
parametric uncertainty. Our analysis reveals that ray-tracing greatly reduces
the signal sensitivity to this uncertainty, contrary to previous calculations
where there was no emission at all in some directions. Based on a
Goldreich-Julian model for the magnetosphere and a Navarro-Frank-White model
for axion density in the galactic centre, we obtain the most robust limits on
the axion-photon coupling, to date. These are comparable to those from the CAST
solar axion experiment in the mass range $\sim 4.2-60\,\mu{\rm eV}$. If the
dark matter density is larger, as might predicted by a "spike" model, the
limits could be much stronger. The dark matter density in the region of the
galactic centre is now the biggest uncertainty in these calculations.
