If the plug goes neither up nor down then it must be - like the Grand Old Duke of York - only half way up.
The plug on my sink falls under it's own weight - if yours does not fall then something is interfering with gravity.
If your plug does not rise then the simple linkage that causes it to rise is defective.
Nobody here can see just what the problem is - it will be a matter of your own observation to make a diagnosis - then the resolution.
Good Luck..., April 2017