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We classify transcendental entire functions that are compositions of apolynomial and the exponential for which all singular values escape on disjointrays. We focus on the case where the escape is degenerate in the sense thatpoints from different singular orbits are arbitrarily close to each other. Asin the general case we employ an iteration procedure on an infinite-dimensionalTeichm\"uller space, analogously to the Thurston's classical TopologicalCharacterization of Rational Functions, but for an infinite set of markedpoints.

Infinite-dimensional Thurston theory and transcendental dynamics III: entire functions with escaping singular orbits in the degenerate case