Parallel algorithms relying on synchronous parallelization libraries often experience adverse performance due to global synchronization barriers. Asynchronous many-task runtimes offer task futurization capabilities that minimize or remove the need for global synchronization barriers. This paper conducts a case study of the multidimensional Fast Fourier Transform to identify which applications will benefit from the asynchronous many-task model. Our basis is the popular FFTW library. We use the asynchronous many-task model HPX and a one-dimensional FFTW backend to implement multiple versions using different HPX features and highlight overheads and pitfalls during migration. Furthermore, we add an HPX threading backend to FFTW. The case study analyzes shared memory scaling properties between our HPX-based parallelization and FFTW with its pthreads, OpenMP, and HPX backends. The case study also compares FFTW's MPI+X backend to a purely HPX-based distributed implementation. The FFT application does not profit from asynchronous task execution. In contrast, enforcing task synchronization results in better cache performance and thus better runtime. Nonetheless, the HPX backend for FFTW is competitive with existing backends. Our distributed HPX implementation based on HPX collectives using MPI parcelport performs similarly to FFTW's MPI+OpenMP. However, the LCI parcelport of HPX accelerated communication up to a factor of 5.