Distilled is not the same as RO/DI as far as the process goes, but ideally both result in the same pure dihydrogen oxide.
I know from experience that pure water will pull stuff effecting the TDS from most plastic storage containers, it's just a matter of time. Food grade plastics take longer and thus are the recommended water storage tanks. My guess is that a bottle of distilled that has sat on the shelves for a year would have a measurable amount of TDS, but I don't know what the leaching rate is of food grade plastics.
Maybe someone with too much time on their hands can buy twelve bottles of distilled water and open one each month to measure the TDS and chart the leaching rate. I don't have that kind of time, nor the desire.
Anyhow, for the purpose of this thread, one gallon jugs of distilled water would likely be the best solution.