  In this paper, we prove the off-shell equation satisfied by the transfer
matrix associated with the XXZ spin-$\frac12$ chain on the segment with two
generic integrable boundaries acting on the Bethe vector. The essential step is
to prove that the expression of the action of a modified creation operator on
the Bethe vector has an off-shell structure which results in an inhomogeneous
term in the eigenvalues and Bethe equations of the corresponding transfer
matrix.
