I took the opportunity of asking flux.ai to design it....
I don't yet understand how this works but here's the overall schematic:
What do you think ? Thanks !

That's because it is AI ideas-in-a-blender, and simply will not work.
You can sort of see what they were trying to do, create a series FET SW, a la SSR's but for that FETS need to be common source, not diodes in series as drawn.
FETS also need level shifters, but negative rails are not common so AI blender missed that detail.
You can do this with a single fet to each rail, and a level shifter to each gate. (PNP or PMOS)
To avoid cross conduction spikes during handover, you can either use 2 diodes, or 2 resistors may be enough at low currents.
What range of actual grid voltages and currents does this need to handle ?
If you need to go to 0V, a bit more care is needed on the FET gate level shifters.