  The $\mathfrak{sl}_2$ weight system, corresponding to the colored Jones
polynomial of knots, is one of the the simplest weight system for chord
diagrams. Recent works have led to explicit computations of this weight system
on chord diagrams with complete and complete bipartite intersection graphs
using $\mathfrak{sl}_2$ weight systems on shares, i.e., on chord diagrams on
two strands.
  In this paper, we continue our study of shares. We prove a conjecture by
Lando about a duality of values of the $\mathfrak{sl}_2$ weight system on chord
diagrams whose intersection graphs are joins of complementary graphs with
discrete ones. To achieve this, we introduce the two-colored intersection graph
of shares, define the inner product of shares, and use chord-adding operators.
