  To determine the topological quantum numbers of fractional quantum Hall (FQH)
states hosting counter-propagating (CP) downstream ($N_d$) and upstream ($N_u$)
edge modes, it is pivotal to study quantized transport both in the presence and
absence of edge mode equilibration. While reaching the non-equilibrated regime
is challenging for charge transport, we target here the thermal Hall
conductance $G_{Q}$, which is purely governed by edge quantum numbers $N_d$ and
$N_u$. Our experimental setup is realized with a hBN encapsulated graphite
gated monolayer graphene device. For temperatures up to $35mK$, our measured
$G_{Q}$ at $\nu = $ 2/3 and 3/5 (with CP modes) match the quantized values of
non-equilibrated regime $(N_d + N_u)\kappa_{0}T$, where $\kappa_{0}T$ is a
quanta of $G_{Q}$. With increasing temperature, $G_{Q}$ decreases and
eventually takes the value of equilibrated regime $|N_d - N_u|\kappa_{0}T$. By
contrast, at $\nu = $1/3 and 2/5 (without CP modes), $G_Q$ remains robustly
quantized at $N_d\kappa_{0}T$ independent of the temperature. Thus, measuring
the quantized values of $G_{Q}$ at two regimes, we determine the edge quantum
numbers, which opens a new route for finding the topological order of exotic
non-Abelian FQH states.
