I don't know if there will be much of a difference in the amount of phased sites, but you should really run per-chromosome or per-genome because of the reasons Geraldine mentioned in the thread Iinked to above.
P.S. It is difficult to tell if the number of phased sites will differ because HaplotypeCaller does physical phasing, which depends on the haplotypes it creates for each active region. You can read more about the active regions and haplotype creation in the Methods section.