I did some testing and would like to report back for the benefit of others looking into these options.
When you plug a Shearwater controller on the OBOE circuit, it shows a small error message in yellow in the same spot where you see SP=0.19 that says "No solenoid" "No O2 sensor". Despite that, everything but the controlling capabilities (changing set points) works, i.e., it monitors and displays the reading from the O2 sensors and all computer functionalities work. Not sure why it says "No O2 sensor" given it can read the sensors.
This means that you can dive with 2 controllers – one plugged to SOLO (this is the one that will fly the unit) and one plugged into OBOE (this will just read the sensors and do the deco calculations). I think this is an interesting option in case you dive in very remote locations and you want to have a controller backup. If you do that, you can as well use it to read the sensors instead of the HUD. The additional benefit is that, in case of main controller failure, you can plug the monitor into the controller and have all the tissue profiles from previous dives.
As a side note, if you disconnect the controller from the SOLO circuit, the electronic will chase whatever setpoint you last programmed with the controller (so you can keep diving in electronic mode even with a main controller failure). Not sure if in this case you would be able to switch the cables underwater in order to start using the backup controller or if you better do it on the surface.