  Semiconic idempotent logic sCI is a common generalization of intuitionistic
logic, semilinear idempotent logic sLI, and in particular relevance logic with
mingle. We establish the projective Beth definability property and the
deductive interpolation property for many extensions of sCI, and identify
extensions where these properties fail. We achieve these results by studying
the (strong) amalgamation property and the epimorphism-surjectivity property
for the corresponding algebraic semantics, viz. semiconic idempotent residuated
lattices. Our study is made possible by the structural decomposition of conic
idempotent models achieved in the prequel, as well as a detailed analysis of
the structure of idempotent residuated chains serving as index sets in this
decomposition. Here we study the latter on two levels: as certain enriched
Galois connections and as enhanced monoidal preorders. Using this, we show that
although conic idempotent residuated lattices do not have the amalgamation
property, the natural class of rigid and conjunctive conic idempotent
residuated lattices has the strong amalgamation property, and thus has
surjective epimorphisms. This extends to the variety generated by rigid and
conjunctive conic idempotent residuated lattices, and we establish the (strong)
amalgamation and epimorphism-surjectivity properties for several important
subvarieties. Using the algebraizability of sCI, this yields the deductive
interpolation property and the projective Beth definability property for the
corresponding substructural logics extending sCI.
