  We construct unitary, stable, and interacting conformal boundary conditions
for a free massless scalar in four dimensions by coupling it to edge modes
living on a boundary. The boundary theories we consider are bosonic and
fermionic QED$_3$ with $N_f$ flavors and a Chern-Simons term at level $k$, in
the large-$N_f$ limit with fixed $k/N_f$. We find that interacting boundary
conditions only exist when $k\neq 0$. To obtain this result we compute the
$\beta$ functions of the classically marginal couplings at the first
non-vanishing order in the large-$N_f$ expansion, and to all orders in $k/N_f$
and in the couplings. To check vacuum stability we also compute the large-$N_f$
effective potential. We compare our results with the the known conformal
bootstrap bounds.
