If I understand what you are saying, you have the clutch depressed to the floor when you start the car but after it starts and you let the clutch pedal out to idle in neutral it starts sinking back towards the floor on its own? Do you have to pull the clutch pedal up with your foot? Does it come back up on its own?
The clutch is hydraulic like brakes so you could have a problem with the master or slave cylinder piston seal not seating properly.
That is so bizaar. I know there is a spring on the pedal arm so it shouldn't be sinking to the floor in the first place unless maybe the spring is missing or popped off. Mine is an auto so can't look at it to see if maybe I could help. Hope you figure it out or someone with knowledge pipes in for you.
Mine does this too. I wouldn't call it sinking so much as just not able to build any pressure on startup. Push the clutch in to start the car, but lifting your foot..it stays down. Won't go into gear either when this happens like the clutch isn't fully disengaged. Definitely a hydraulic issue between clutch master and slave cylinders.