They almost certainly will; that's just ECON101. I'm sure H&M and Zara aren't just sitting around admiring themselves in the mirror.
Clothing is a weird market but there is definitely an elastic low-end and Uniqlo is playing in that space. There's no reason to believe the rules of basic economics are not functioning.
With that said, what do you think happens when a second company in the same sector starts to realize the same cost savings, and realizes they can move more units by using the cost savings to lower prices and ultimately return even more to shareholders?
Again, you're completely right. Companies don't lower prices out of the goodness of their cold, black, capitalist hearts. Yet, might there be other factors at work here?