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?

Note that these two are entirely compatible. Prices routinely go down and customers receive benefit but they do not notice.

For example, I wrote about research on effect of barcode scanner here: https://news.ycombinator.com/item?id=11414435

Can you name a single instance where prices did _not_ go down due to automation? You're saying process won't go down tomorrow, but of course that's not how any of this works.

Automation is correlated with cost savings for consumers because usually the competition can figure out the same thing after seeing another player do it. It's a mistake to think they are causally related

Prices also go down if it will lead to more profit. If you can sell 2.1 times as many widgets selling at half the profit then of course you will.

Well sure, but in what world is that going to happen in here? They didn't invent time travel; they automated a factory. This can be replicated. They may also lower prices simply to undercut competitors if they think the increase in volume will work out to more profits overall.