Which of the following statements is FALSE for path compression in union$-$find data structure?

A: Path compression with the "simple version" of Union$-$Find implementation improves the worse$-$case complexity of find$()$ from log$($N$)$ to inverse Ackermann function, which is much slower than log$($N$)$

B: Inverse Ackermann function grows extremely slowly, and is even slower compared to log$($N$)$

C: Path compression improves the amortized complexity from O$($m log$($N$))$ to O$($m$$Inv$.$Ackermann$($m$,$N$))$ for m consecutive operations and N elements

D: Path compression can be easily implemented with simple modification in the standard find$($x$)$ function.