I don't think anybody has a 100% right answer. Unless a whole lot more research by a non interested (financially) party is involved.
So we can only tell by our personal experience.
Softer carpet due to cleaning with soft water is most likely due to less mineral in water. Does soft mean cleaner and stay clean longer? In my opinion not necessarily.
Will a reasonably diluted detergent rinse that was properly formulated add to faster re soiling?- In my experience tested on hundreds of home- It does not.
Most detergent formulations are designed to work throughout the country. In a sense they are a compromise, as different parts of the country have very different water characteristics. So they try to design a product that will work everywhere. Most contain water softeners in them. So, not considering what happens to the plumbing of our machines, is soft water really necessary for cleaning purposes, if water softeners are formulated into the detergent?
Water is softened (if I am not wrong) by increasing salt content and the ability of the water to contain (attach, not the correct scientific terminology) to minerals. So when cleaning with soft water, we are leaving less minerals but the water contains more salt. Is the effect of the salt left in the carpet any different than the detergent? Better? Worse?
Also there's the law of diminishing returns. Let us say that by using these processes, we'll achieve a 5% improvement in re soiling. Is it worth it? What other effects do water softening or ionizing have on carpet durability, dye stability, re soiling, etc. does it have?
In my personal experience, using soft water, acid rinses and just water rinse, I did not see any improvement in looks, long term cleanliness and lower re soiling rate, in comparison to a mildly diluted detergent rinse. No improvement what so ever. The improvement that I did see when using the detergent rinse was: Less call backs (statistically insignificant) as I get very few, less returning stains and definitely substantial time saving in the cleaning process.