This seems a common problem among E39s; just google it and you will see how many search results you will get. Hopefully yours won't have this problem.
Just some updates:
The shake problem is also apparent at low speeds 30km/h and below. You know those very slow rush hours times? The car is like crippled or something! It's almost like the diagonal wheels are one size and the other diagonal wheels are another size! Very strange characteristic!
I still haven't taken the car to RMP Motors. Planning to do as soon as possible.