Obviously. the 119 armor is an entirely different item from the 113 item, and thus will not have the same augments as an entirely different item. This is what you get with item level systems, nothing you work on matters for a more than a month, because you'll have to toss it when you get something better in the next event.

Of course, had we leveled up through an extended merit system, or had we just increased our actual levels, our work wouldn't be wasted, because our strength would come from our characters getting stronger rather than our gear. That and you would have to take all 22 jobs to level 119, instead of now, where you take BRD to 119, and use it to get the gear for the other 21, instantly jumping them to 119 at the same time.