Zip your Lip
To stop talking, to remain secret or calm
Earlier this idiom was used in 1868 as ‘button your lips’ because during that period of time buttons were used to close things. After Zippers started being used this idiom was modified to ‘zip your lip’ by the people. The use of this idiom can be traced back to 1943.
And then the politician asked them all to zip their lips as he wanted to state his point first.