My Ls430 2002 has the same problem, suspension going down on passenger side. Has got progressively worse, but come up to normal when I turn on ignition and the compressor kicks in.
I see from above this was repaired for £175 and £75 fitting, I see the cost oF a new shock is about a grand. So was interested to know what you had replaced.
I had to have my wheel bearing on this wheel replaced recently, the garage said it wasn't related, but I'm not sure he is correct