The cryptographic sponge is a popular method for hash function design. The construction is in the ideal permutation model proven to be indiifferentiable from a random oracle up to the birthday bound in the capacity of the sponge. This result in particular implies that, as long as the attack complexity does not exceed this bound, the sponge construction achieves a comparable level of collision, preimage, and second preimage resistance as a random oracle. We investigate these state-of-the-art bounds in detail, and observe that while the collision and second preimage security bounds are tight, the preimage bound is not tight. We derive an improved and tight preimage security bound for the cryptographic sponge construction. The result has direct implications for various lightweight cryptographic hash functions. For example, the NIST Lightweight Cryptography finalist Ascon-Hash does not generically achieve 2¹²⁸ preimage security as claimed, but even 2192 preimage security. Comparable improvements are obtained for the modes of Spongent, PHOTON, ACE, and Subterranean 2.0, among others.