Before revision:  In this work, we provide a simple  monotone CR scheme with a balancedness factor of 1 - e^{-k}k^k/k! for uniform matroids of rank k  , and show that this is optimal.
After revision:  In this work, we provide a simple monotone CR scheme with a balancedness factor of 1 - e^{-k}k^k/k! for uniform matroids of rank k (which matches the balancedness of 1-1/e for k=1) , and show that this is optimal.