  The fractional quantum Hall (FQH) effect at the filling number $\nu=5/2$ is a
primary candidate for non-Abelian topological order, while the fate of such a
state in the presence of random disorder has not been resolved. Here, we
address this open question by implementing unbiased diagnosis based on
numerical exact diagonalization. We calculate the disorder averaged Hall
conductance and the associated statistical distribution of the topological
invariant Chern number, which unambiguously characterize the disorder-driven
collapse of the FQH state. As the disorder strength increases towards a
critical value, a continuous phase transition is detected based on the disorder
configuration averaged wave function fidelity and the entanglement entropy. In
the strong disorder regime, we identify a composite Fermi liquid (CFL) phase
with fluctuating Chern numbers, in striking contrast to the well-known
$\nu=1/3$ case where an Anderson insulator appears. Interestingly, the lowest
Landau level projected local density profile, the wavefunction overlap, and the
entanglement entropy as a function of disorder strength simultaneously signal
an intermediate phase, which may be relevant to the recent proposal of
Pfaffian-anti-Pfaffian puddle state.
