  We construct integral representations of solutions to the boundary quantum
Knizhnik-Zamolodchikov equations. These are difference equations taking values
in tensor products of Verma modules of quantum affine $\mathfrak{sl}_2$, with
the K-operators acting diagonally. The integrands in question are products of
scalar-valued elliptic weight functions with vector-valued trigonometric weight
functions (boundary Bethe vectors). These integrals give rise to a basis of
solutions of the boundary qKZ equations over the field of quasi-constant
meromorphic functions in weight subspaces of the tensor product.
