Attempts to test the anomaly using other spacecraft such as Galileo and the Voyager probes have proved unsuccessful,
First off, let's be clear what these guys are talking about: the trajectory determination process has shown what appears to be an unexplained perturbation. Apparently the same perturbation has not been noted with Voyager.
Note that the data in question here are tracking data gathered using Very Long Baseline Interferometry.
While it's entirely possible that they're observing some real effect, the apparent lack of corroborating data for other deep space probes suggests to me that there's a subtle error somewhere in the ground system -- probably in the databases used for Pioneer trajectory processing.
It is not clear why the authors have tossed Galileo into this -- it was a Jupiter orbiter, far closer to the sun than the Pioneers and Voyagers, and its perturbation environment was much different. The effect, if real, is being detected outside the solar system.
My bet: it's something stupid and overlooked on the ground. It almost always is.
I have a mental picture of an "ice-rimed" and ancient Mariner coasting slowly along. The gradual accumulation of interstellar frost would be something they have no way to measure.
The overall mass would obey the same gravitational rules, but the accumulating frost would be at a slower velocity, gradually slowing the craft in an inexplicable manner.