  We classify transcendental entire functions that are compositions of a
polynomial and the exponential for which all singular values escape on disjoint
rays. We focus on the case where the escape is degenerate in the sense that
points from different singular orbits are arbitrarily close to each other. As
in the general case we employ an iteration procedure on an infinite-dimensional
Teichm\"uller space, analogously to the Thurston's classical Topological
Characterization of Rational Functions, but for an infinite set of marked
points.
