The secret-key rate measures the rate at which Alice and Bob can extract secret bits from sampling a joint probability distribution, unknown to an eavesdropper Eve. The secret-key rate has been bounded above by the intrinsic information and reduced intrinsic information. However, we prove that the reduced intrinsic information is 0 if and only if the intrinsic information is 0. This result implies that at least one of the following two conjectures is false: bound secrecy exists, or the reduced intrinsic information equals the secret-key rate. We give an explicit construction of an information-erasing binarization for a candidate for bound secrecy. We then introduce some approaches for proving the existence of bound secrecy, such as reducing the channel space, linearly transforming Bob's map, and perturbing a channel for Eve.