I've given some thought to this, I agree it's one of the obvious improvements. I think with a refractive design like this one it's difficult, because of the way a lens works. Most other projects like this including commercial ones like CoeLux use a reflective design, and I believe that's partly for this reason.

I’ve seen the video he links at the top of the article, a few years back. They took an old satellite dish, silvered the surface, and put a giant LED where the receiver had been so the light produced was nearly parallel. Otherwise the shadows look wrong.

In his case he sacrificed a doorway or a closet to make a window with “real light”

Would moving the box itself help provide that effect? Maybe put it on a track to move throughout the day?

There was a project posted here awhile ago with something similar, but the light was larger and he positioned it outside a fake window with a curtain across it.