That nailed it! Having the mesh as a single asset made it very reproducible.
I found a couple of issues when trying this. Using a similar setup to your scene with three levels would cause only a single reflection plane to work at a time. However, using a box inset and having reflective faces inside would do something similar if I had a reflection plane for all interior faces. They would work in Editor but in PIE or using Game View (hotkey G) would only use 2 of them out of the 4 I placed. Certainly something worth investigating and thanks for your report!
I’ve submitted a ticket for this that you can track here: Unreal Engine Issues and Bug Tracker (UE-38140)
Thanks!