Ok, I understand now. Thank you.
Including
captures the practitioner's heuristic "guess" that hypothesis that are closer to zero are "simpler". So the check is, very loosely, that the above assumption is true. Making all the features have zero mean is probably sufficient (in many applications) for the assumption to be "reasonable". However, it is not strictly related since the assumption could hold (or not) for other reasons depending on the application.
In either case, performing validation should allow us to narrow down the types of regularizers that make sense for a particular data set and application.