  In this paper we estimate from above the area of the graph of a singular map
$u$ taking a disk to three vectors, the vertices of a triangle, and jumping
along three $\mathcal{C}^2-$ embedded curves that meet transversely at only one
point of the disk. We show that the relaxed area can be estimated from above by
the solution of a Plateau-type problem involving three entangled nonparametric
area-minimizing surfaces. The idea is to "fill the hole" in the graph of the
singular map with a sequence of approximating smooth two-codimensional surfaces
of graph-type, by imagining three minimal surfaces, placed vertically over the
jump of $u$, coupled together via a triple point in the target triangle. Such a
construction depends on the choice of a target triple point, and on a
connection passing through it, which dictate the boundary condition for the
three minimal surfaces. We show that the singular part of the relaxed area of
$u$ cannot be larger than what we obtain by minimizing over all possible target
triple points and all corresponding connections.
