  We apply the Grassmannian representation of the dimer model, an equivalent
approach to Kasteleyn's solution to the close-packed dimer problem, to
calculate the connection probabilities for the double-dimer model with
wired/free/wired/free boundary conditions, on a rectangular subdomain of the
square lattice with four marked boundary points at the corners. Using some
series identities related to Schwarz-Christoffel transformations, we show that
the continuum of the result is consistent with the corresponding one in the
upper half-plane (previously obtained by Kenyon-Wilson), which is in turn
identical to the connection probabilities for 4SLE$_4$ emanating from the
boundary, or equivalently, to a conditioned version of CLE$_4$ with
wired/free/wired/free boundary conditions in the context of conformal loop
ensembles.
