Motivated by the experimental observation of a quantized 5/2 thermal conductance at filling nu = 5/2, a result incompatible with both the Pfaffian and the anti-Pfaffian states, we have pushed the expansion of the effective Hamiltonian of the 5/2-quantized Hall state to third order in the parameter K = EC/h over bar wc oc 1/./B controlling the Landau-level mixing, where EC is the Coulomb energy and wc is the cyclotron frequency. Exact diagonalizations of this effective Hamiltonian show that the difference in overlap with the Pfaffian state and the anti-Pfaffian state induced at second order is reduced by third-order corrections and disappears around K = 0.4, suggesting that these states are much closer in energy at smaller magnetic field than previously anticipated. Furthermore, we show that in this range of K the finite-size spectrum is typical of a quantum phase transition, with a strong reduction of the energy gap and with level crossings between excited states. These results point to the possibility of a quantum phase transition at smaller magnetic field into a phase with an emergent particle-hole symmetry that would explain the measured 5/2 thermal conductance of the 5/2-quantized Hall state.