Can the gravitational potential in a spiral galaxy be positive?
I think your recreation is essentially correct. For example, if you look at Figure 1 in that paper, you can see that the potential goes from positive to negative as a function of the azimuthal angle ("phase").

What you're missing is that the potential and density functions they define are perturbations, which are intended to be added to an axisymmetric galaxy model. The idea is that the axisymmetric disk is modulated by their perturbation, so that the total density is less than average (but not less than zero!) where their perturbation is $&lt; 0$ and greater than average where their perturbation is $&gt; 0$. (Similarly, the total potential is $&lt; 0$ everywhere; in the regions where their perturbation is positive, the total potential becomes less negative, but never $&gt; 0$.)

As they point out (pp.4-5), "The impressions invoked by the density distributions of Figures 3 and 5 can be somewhat misleading. These densities must be regarded as perturbations to an azimuthally uniform stellar disk with the same vertical scale height."

And: "In Figure 10, a disk component with the same radial dropoff and scale height as the perturbation, and just sufficient amplitude to make the net density everywhere positive, has been added to the perturbation density." and "In Figures 12 and 13, various amounts of perturbation density are shown added to a representative full stellar disk."