To continue a bit..
The property of changing dimensionality is an emergent feature of spacetime for Causal Dynamical Triangulations and Quantum Einstein gravity theory, as is mentioned by the authors. But the idea of a spectral dimension that runs from 2-d near the Planck scale to 4-d as we approach the common scale is attractive to a number of Quantum Gravity theories now under serious consideration, including loops, spin foams, and various theories of causal structure. To constructivist geometers, there is no dimensionality until a means of measurement is constructed, which gives determination a dual face. That is; determination by measurement is construction or creation - at its core.
However at this point; I wonder whether what is most realistic is some sort of bi-metric approach, where there exists an upper and lower bound to what can properly be referred to as dimensionality. I note here that String theory and Supergravity operate in a 10-d or 11-d space respectively, near the Planck scale, and I do not find this to be in conflict with the lower bound of 2-d described by CDT and QEG - so long as a bi-metric cosmos is allowed. But I also wonder whether our spacetime is evolving past the 4-d phase. The fact that the 5-d sphere offers maximal hypervolume would seem to suggest that the end of dimensional evolution would be to a cosmos of five dimensions.
Intriguingly; the authors of this paper note that their result is at odds with most versions of the inflationary universe scenario. However; this may make for a closer reflection of the Planck satellite results, and other recent data, in future cosmological models. It would seem Steinhardt was right to call his own Inflationary model into question, but it remains to be seen which proposed model is the most appropriate replacement.
All the Best,
Jonathan