I really don't think so. For one thing, they don't know about the mind control, so from their perspective, the odds of that being obeyed are really, really low.
For another, at least Wookieepedia claims that the general orders could be activated by the chancellor, or the republic security council, with no mention of anyone else. Which makes sense, Palpatine hardly wants anyone else to be able to activate any of the other orders just by saying so (I mean, at least in Legends, one of those orders is 'kill the Chancellor, he's a traitor.' Again, this is from Legends, but there Order 66 expressly is stated as:
"In the event of Jedi officers acting against the interests of the Republic, and after receiving specific orders verified as coming directly from the Supreme Commander (Chancellor), GAR commanders will remove those officers by lethal force, and command of the GAR will revert to the Supreme Commander (Chancellor) until a new command structure is established."
Which makes a lot of sense. Palpatine's not real big on either information, or power sharing.
To the extent they even know they exist, they presumably lack the authority to trigger the mind control part of the plan, which is sort of critical to it actually working.