  The open XXZ spin chain with the anisotropy parameter $\Delta=-\frac12$ and
diagonal boundary magnetic fields that depend on a parameter $x$ is studied.
For real $x>0$, the exact finite-size ground-state eigenvalue of the spin-chain
Hamiltonian is explicitly computed. In a suitable normalisation, the
ground-state components are characterised as polynomials in $x$ with integer
coefficients. Linear sum rules and special components of this eigenvector are
explicitly computed in terms of determinant formulas. These results follow from
the construction of a contour-integral solution to the boundary quantum
Knizhnik-Zamolodchikov equations associated with the $R$-matrix and diagonal
$K$-matrices of the six-vertex model. A relation between this solution and a
weighted enumeration of totally-symmetric alternating sign matrices is
conjectured.
