  In this paper, which is the last of a series of three, we first verify that
the two open-closed effective potentials derived in the previous paper from the
WZW theory in the large Hilbert space and the $A_\infty$ theory in the small
Hilbert space have the same vacuum structure. In particular, we show that
mass-term deformations given by the effective (two open)-(one closed) couplings
are the same, provided the effective tadpole is vanishing to first order in the
closed string deformation. We show that this condition is always realized when
the worldsheet BCFT enjoys a global ${\cal N}=2$ superconformal symmetry and
the deforming closed string belongs to the chiral ring in both the holomorphic
and anti-holomorphic sector. In this case it is possible to explicitly evaluate
the mass deformation by localizing the SFT Feynman diagrams to the boundary of
world-sheet moduli space, reducing the amplitude to a simple open string
two-point function. As a non-trivial check of our construction we couple a
constant Kalb-Ramond closed string state to the OSFT on the
$\text{D}3$--$\text{D}(-1)$ system and we show that half of the bosonic
blowing-up moduli become tachyonic, making the system condense to a bound state
whose binding energy we compute exactly to second order in the closed string
deformation, finding agreement with the literature.
