  A recent thermal Hall conductance experiment [Banerjee et al., Nature
{\bf559}, 205 (2018)] for $\nu = 5/2$ fractional quantum Hall system appears to
rule out both the Pfaffian and anti-Pfaffian and be in favor of the PH-Pfaffian
topological order, while the existing numerical results without disorder have
shown otherwise. In this paper we offer a possible resolution by proposing a
new state, termed compressed PH-Pfaffian state by "compressing" the PH-Pfaffian
state with two flux quanta removed to create two abelian Laughlin type
quasiparticles of the maximum avoidance from one another (or of the maximum
number of zeros). The compressed PH-Pfaffian state is not particle-hole
symmetric but possesses the PH-Pfaffian topological order. In spherical
geometry, the compressed PH-Pfaffian state has the same magnetic flux number
$N_{\phi}= 2N-3$ as the Pfaffian state, allowing a direct numerical comparison
between the two states. Results of exact diagonalization of finite
disorder-free systems in the second Landau level show that, by increasing the
short range component of the Coulomb interaction, the ground state undergoes a
phase transition from the Pfaffian state to the compressed PH-Pfaffian state
before further entering into a gapless state. The low energy gapped excited
states result from the breakup of the abelian Laughlin type quasiparticle into
two non-abelian quasiparticles.
