Why would the interplanetary medium atomic hydrogen Lyman‐α emission fall as 1/r rather than1/r^2?
The brightness considered here is the brightness of scattered Lyman $\alpha$ as viewed in the upstream direction, i.e. away from the Sun.

The authors assume that the density of the atoms responsible for scattering the Ly$\alpha$ — the neutral hydrogen atoms — is 1) roughly constant throughout the Solar System and 2) is optically thin, meaning that the radiation field is not attenuated considerably with distance from the Sun. 

Since the intensity of a point source falls off as $1/r^2$, the scattering emissivity in a given point should then also fall off roughly as $1/r^2$.

But when viewed from inside the Solar System and in the upstream direction, the brightness is the integral of all scattering events from that point to infinity, and since $\int 1/r^2 \propto 1/r$, you have your quoted brightness.