  We prove a strengthening of the "reciprocity conjecture" of Khare and
Wintenberger. The input to the original conjecture is an odd prime p, a CM
number field F containing the pth roots of unity, and a pair of primes of the
maximal totally real subfield E of F that are inert in the cyclotomic
Z_p-extension of E. In analogy to a statement about generalized Jacobians of
curves, the conjecture asserts the equality of two procyclic subgroups of the
Galois group G of the maximal p-ramified abelian pro-p extension of E. The
first is the intersection of the subgroup of G fixing the cyclotomic
Z_p-extension with the closed subgroup of G generated by the two primes. The
second is generated by the class of an exact sequence defining the minus part
of the p-part of the ray class group of the cyclotomic Z_p-extension of F of
conductor the product of the two primes of E.
