I suppose it can tell the difference between RUN and ACC on your key switch. It stays on when it sees ACC, and only shuts off when it sees OFF.
Your car may also have the OBD2 protocol that puts the SG to sleep when the engine is killed with the key going from RUN to ACC (or through a kill switch wired in the key circuit effectively doing the same thing). If that's the case case, Hybrid mode won't help.
But try Hybrid mode first and see if the SG stays awake when you kill switch the engine.
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