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.
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