Main bijection conjecture for physical Bethe solutions and rigged configurations

Let physical solutions of the Bethe ansatz equations be decomposed into strings, let $\nu$ be the partition formed by their string lengths, and let the riggings be assigned by the string-crossing procedure described in the source using modified Bethe numbers and vacancy numbers. Main bijection conjecture. The above procedure gives a one-to-one correspondence between the set of physical solutions to the Bethe ansatz equations and the rigged configurations. This is the paper's proposed explicit construction of the Bethe-solution/rigged-configuration bijection; the supplied text gives no evidence of its resolution.