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We establish sufficient conditions which ensure that a locally-warped productstructure propagates backward in time under the Ricci flow. As an application,we prove that if an asymptotically conical gradient shrinking soliton isasymptotic to a cone whose cross-section is a product of Einstein manifolds,the soliton must itself be a multiply-warped product over the same manifolds.

Backward propagation of warped product structures and asymptotically conical shrinkers