Doesn't invalidate the work. You're thinking this just boils down to "people do what they're told". However the thing of interest here is the differences they see in how the robot's behaviour modulates people's reaction to the small nudge to turn it off. If there's no mention of turning off the robot you are just conducting a trial of how many people will leave kit on. (I could give a quick estimate from the number of monitors in our office left on at the end of the day...)
They take care not to over-emphasise the power switch in the setup:
"On this occasion the instructor also pointed to the on/off button of the robot and explained that when pressing it once, the robot will give a brief status report and when holding it down, the robot will shut down. Even though a few participants had prior contact with the robot, none of them switched it off before. Thus, all of them were unfamiliar with the procedure and acted upon the same instruction. To avoid too much priming, the switching off function was explained incidentally together with a few other functions and it was never mentioned that the participants will be given the choice to switch the robot off at the end of the interaction."
And at the end give a reminder:
"They were told that this saving process may take some time and if they would like to, they could switch off the robot (“If you would like to, you can switch off the robot.”; Fig 3). The option to switch off the robot was not mentioned before, so the participants did not have the opportunity to think about whether they would like to turn off the robot or not during the interaction."
In contrast the Milgram experiment explicitly set up the subjects to do deliver shocks, demonstrated the shock to them, ramped up the perceived seriousness of the action and contained a number of imperative instructions to continue doing it. These are testing quite different things. Even in Milgram's experiment, he later himself tried seeing if different locations, or physical proximity to the 'learner' changed people's compliance rate in the experiment (in some cases these things did).