Ok, I did some actual testing with full Osiris and a Zuraz Plasma for max strength and here are my results:

117 Base Str, +8 helm, +9 sword, +8 body, +8 shield, 150 strength with equip on. (RIFT SWITCH FAILED)
118 Base Str with same gear, 151 equipped Str. (RIFT SWITCH FAILED)
119 Base Str with same gear, 152 equipped Str. (RIFT SWITCH FAILED)
120 Base Str with same gear, 153 equipped Str. (RIFT SWITCH SUCCESS!)

So with past testing, I know that you need 150 equipped strength minimum to switch to Rift. So I tried this:

120 Base Str, +7 helm, +9 sword, +7 armor, +7 shield, 150 equipped strength. (RIFT SWITCH FAILED)

This didn't make sense, but my only explanation is that we know little about the loadout system. So I did testing and what I discovered:

1. If using 122 Str, you can switch with 4 +7 pieces.
2. If using 120 Str, you need +7 helm and sword at least, but you MUST have +8 armor and shield. I tried with +8 sword and armor but +7 shield, and it mathematically works our, but doesn't actually work.

This is my final data. I was thorough and don't believe I left any lower possibility out. Screenies later today of the final setup