The OP stated that some guys think if the light has gone out they don't have to shut the engine down. That's "making up stuff".
Summary - shut it down, unless some highly unlikely to occur event happened that makes delaying the burning engine shutdown the prudent course of action.
Odds of that exception happening? Small, but it's amazing to watch guys start to run down the "maybe's it's this" trail. early in the trouble shooting.
