tl;dr: The InSight lander, which touched down in Nov'18 has found that the planet's magnetic field sometimes starts to pulsate in ways that have never before been observed, e.g usually at local midnight. The cause is currently unknown.
The fact that the phenomenon is associated with local midnight implies that it is not a geophysical effect, but connect to the interaction of the solar wind with the planetary magnetic field.
This is also what the paper you linked said: "We speculate that the observed magnetic pulsations by InSight to date are associated with fluctuations in the induced magnetotail and on the magnetospheric boundary."
So unless a researcher has a bright idea the next step is to run high resolution 3d hybrid-kinetic simulations of the Martian magnetosphere and play with the upstream solar wind conditions to see if they can make the oscillations show up in simulations. But that is not exactly easy or (computationally) cheap. But if we are lucky there will be new insights in time for the AGU meeting in December.
