  We numerically address the issue of which monopole operators are relevant
under renormalization group flow in three-dimensional parity-invariant
noncompact QED with $4$ flavors of massless two-component Dirac fermion. Using
lattice simulation and finite-size scaling analysis of the free energy to
introduce monopole-antimonopole pairs in $N=4$ and $N=12$ flavor noncompact
QED$_3$, we estimate the infrared scaling dimensions of monopole operators that
introduce $2\pi$ and $4\pi$ fluxes around them. We first show that the
estimates for the monopole scaling dimensions are consistent with the large-$N$
expectations for $N=12$ QED$_3$. Applying the same procedure in $N=4$ QED$_3$,
we estimate the scaling dimension of $4\pi$ flux monopole operator to be
$3.7(3)$, which allows the possibility of the operator being irrelevant. This
finding offers support to the scenario in which higher-flux monopoles are
irrelevant deformations to the Dirac spin liquid phase that could be realized
on certain non-bipartite lattices by forbidding $2\pi$-flux monopoles.
