  In the context of boundary conformal field theory, we investigate whether the
boundary trace anomaly can depend on marginal directions in the presence of
supersymmetry. Recently, it was found that a graphene-like non-supersymmetric
conformal field theory with a four-dimensional bulk photon and a
three-dimensional boundary electron has two boundary central charges that
depend on an exactly marginal direction, namely the gauge coupling. In this
work, we supersymmetrize this theory, paying special attention to the boundary
terms required by supersymmetry. We study models with 4, 8, and 16 Poincar\'e
supercharges in the bulk, half of which are broken by the boundary. In all
cases, we find that at all orders in perturbation theory, the gauge coupling is
not renormalized, providing strong evidence that these theories are boundary
conformal field theories. Moreover, the boundary central charges depend on the
coupling. One possible exception to this dependence on marginal directions is
that the difference between the two charges is coupling independent at one-loop
in the maximally supersymmetric case. In our analysis, a possible boundary
Chern-Simons term is incorporated by a bulk $\theta$-term.
