Daschner 1996 gave a look into ion exchange filters - it was devastating: "In 4 of 6 filters tested in the laboratory, bacterial counts in the fresh filtrate were higher than in tap water after approximately one week of use both at room temperature and at 4°C, suggesting growth or biofilm formation in the filter material. In some cases colony counts in the filtered water were 10,000 times those in tap water. The filter material of 5 of 13 new commercial filters was contaminated with bacteria or moulds."
As Zero water only makes claims about he removal of stuff ("dissolved solids" - aka the stuff that we pay for in mineral water), the (re-)introduction of stuff is (wisely) not covered in their marketing material. As the ion-exchange filter will work fine on chlorine, if the water was devoid of bacteria before, and safe against the introduction of microorganisms via added chlorine, after filtering the water will have added bacteria and fungi, and no chlorine to counteract. Cool trade.