Thinking of the void as an analogy to a gas is misleading, that's why your supposition doesn't work.
The universe is expanding, everywhere, as far as we can tell. Without any evidence to the contrary (and a fair bit of evidence supporting it), the expansion is constant, everywhere, and is described by a value called, you guessed it, the Hubble Constant. Most recently, however, a handful of different measurements have suggested two disparate values for the Hubble Constant, resulting in what's known as the Hubble Tension ("tension" because the two values appear to be being measured correctly, but are not the same).
The idea in this paper, as far as I understand it, is that there are large-scale fluctuations in mass distribution, and we happen to be in a local minimum compared to the average across the universe, thus a local void. Since the Universe isn't filled with a gas under pressure (at least to first approximation), there isn't a grand rushing-in to fill that void. Rather the opposite: the relatively higher density elsewhere is pulling harder by its larger gravity than the mass in our local neighborhood, creating a local anomaly in the large-scale gravitational field, pulling us outward. This local anomaly appears as a local increase to the Hubble constant, adding on to the underlying Hubble expansion. We are, locally, expanding slightly faster than the Universal average.
This local effect might explain the two measurements for the Hubble Constant, one in our region of the Universe which is affected by the local paucity of mass, and one across greater expanses of the Universe where the lumpy distribution of mass is evened out by the law of large numbers.