I am trying to understand a typical explanation for the need of a dark matter halo in galaxies. The rotational velocity of stars around the centre of a galaxy seems to be constant past a certain radius. Assuming we can use
$v(r)=\sqrt\frac{GM(r)}{r}$,
where $M(r)$ is the mass of the galaxy enclosed within radius $r$, we seem to need $M(r)\propto r$ for the velocity to be constant. Assuming we can use $M(r)=\frac{4}{3}\pi r^3\rho(r)$, this requires $\rho(r) \propto 1/r^2$. Below is an image of the surface density profile of galaxy NGC 3198. Let's say the volume density profile looks similar and it can be modelled as $1/r^2$. This would make the constant velocity behaviour plausible without the need for dark matter.
The problem arises from the fact that the visible matter (stars, gas) in galaxies have a cut-off radius $R_{\textrm{visible}}$ past which there is virtually none. We do not receive any luminosity from regions located at $r>R_{\textrm{visible}}$. The claim is then that the velocity curve is still constant past $R_{\textrm{visible}}$, and hence the mass of the galaxy has to keep increasing linearly with radius. This extra invisible mass leads to postulate that galaxies are embedded in a dark matter halo of radius $R_{\textrm{dark}}>R_{\textrm{visible}}$.
My question is, how can we know that the velocity curve is still constant for $r>R_{\textrm{visible}}$ if we cannot detect anything coming from such outer region?