Slavnov and Gaudin-Korepin formulas for models without $U(1)$ symmetry: the XXX chain on the segment

We consider the isotropic spin$-\frac{1}{2}$ Heisenberg chain with the most general integrable boundaries. The scalar product between the on-shell Bethe vector and its off-shell dual, obtained by means of the modified algebraic Bethe ansatz, is given by a modified Slavnov formula. The corresponding Gaudin-Korepin formula, $\mathit{\text{i.e.}}$, the square of the norm, is also obtained.