Very good observation. I didn't pay attention to nesting. It didn't even occur to me that this interpretation of the sentence was also possible. (And as far as I can tell it's is a perfectly valid interpretation too!)

It all comes down to the precedence levels of the LID particles (に, は, で etc.), doesn't it. I'm not sure if those are well-defined, but I think it would feel more natural for a native speaker to evaluate に before は in nearly all cases. The order between で and は seems more elusive. I'll definitely bring this up in our conversation group next time, though I'm sure I'll get some funny looks for it.