I had a question a several days ago about the publicVariable command that no one helped me to answer. However, in my research about the command, I think it'll do exactly what you need.
If I'm not mistaken, you use publicVariable everytime you assign a new value to a global variable, which will then broadcast the variable's new value to all clients. So, relating to your code, immediately after changing the value to false for your variable, put the following code:
publicVariable varName
Where varName is the name of your variable.
Hope it works. Good luck.