It does not stay closed because junk is blocking it.
You need to remove /clean and grease it.
It is easy and uses just a flat head screwdriver and a Phillips ,both located in your Lexus tool kit.
Put the car in neutral or 4/d , engine off.
Close the cup holder as much as possible ( use a helper to keep it closed in the other seat).
Open the lower armrest then place the flat screwdriver and lift up on the wood close to the cup holder on the vinyl. Do both side of the wood to pop up. The lift the wood and pull towards the rear of the car to release the other 2 clips by the ashtray. Move it to the side to expose the underneath.
Here is where you want to have a vacuum and Windex as it will be dirty under there.
Undo the 4 Phillips screws and lift up the cup holder . Clean it and remove the junk blocking it.
Luckily you asked today so it is fresh in my mind.
I just did my friends car today and it had some unspeakable things in there including a Bluetooth earpiece he lost.