As one big, falsifiable scientific hypothesis... not well. It wasn't all that strong to begin with, in this sense.

As a collection of fascinating ideas about the development of modern human consciousness... fantastically well. Modern ideas from all over the place: science fiction, human history, philosophy, spiritualism and even computer science have broadly moved closer to Jaynes' way of thinking.

One exception is his definition of consciousness. We still don't really have a single definition, and every "theory" tends to define it different. But overall, Jaynes' doesn't mesh too well with most current definitions. We're much more likely to consider animals conscious today, for example. You might substitute "introspective consciousness."

In particular, modern notions that "consciousness is a simulation/projection" work well with Bicameral Mind.