  We study the IR phase of three dimensional quantum electrodynamics (QED$_3$)
coupled to $N_f=2$ flavors of two-component Dirac fermions, which has been
controversial for decades. This theory has been proposed to be self-dual with
symmetry enhancement $(SU(2)_f\times U(1)_t )/\mathbb{Z}_2\rightarrow O(4)$ at
the IR fixed point. We focus on the four-point correlator of monopole operators
with unit topological charge of $U(1)_t$. We illustrate the $O(4)\rightarrow
SU(2)_f\times U(1)_t $ branching rules based on an $O(4)$ symmetric positive
structure in the monopole four-point crossing equations. We use conformal
bootstrap method to derive nonperturbative constraints on the CFT data and test
the conformality and self-duality of $N_f=2$ QED$_3$. In particular we find the
CFT data obtained from previous lattice simulations can be ruled out by
introducing irrelevant assumptions in the spectrum, indicating the IR phase of
$N_f=2$ QED$_3$ is not conformal.
