  In a recent work, Maciej Do\l{}e\k{}ga and the author have given a formula of
the expansion of the Jack polynomial $J^{(\alpha)}_\lambda$ in the power-sum
basis as a non-orientability generating series of bipartite maps whose edges
are decorated with the boxes of the partition $\lambda$.
  We conjecture here a variant of this expansion in which we restrict the sum
on maps whose edges are injectively decorated by the boxes of $\lambda$. We
prove this conjecture for Jack polynomials indexed by 2-column partitions. The
proof uses a mix of combinatorial methods and differential operator
computations.
