  The holographic complexity of formation for the AdS$_3$ $2$-sided
Randall-Sundrum model and the AdS$_3$/BCFT$_2$ models is logarithmically
divergent according to the volume conjecture, while it is finite using the
action proposal. One might be tempted to conclude that the UV divergences of
the volume and action conjectures are always different for defects and
boundaries in two-dimensional conformal field theories. We show that this is
not the case. In fact, in Janus AdS$_3$ we find that both volume and action
proposals provide the same kind of logarithmic divergences.
