In a recent paper with Daniel J. Bernstein, we introduce a new security notion for cryptographic hash functions called decisional second-preimage resistance (DSPR) that asks to decide if a given domain element has a colliding value within the domain of the hash function. It turns out that DSPR and conventional second-preimage resistance (SPR) together imply one-wayness (PRE). This fills a gap in the theory of hash functions showing when SPR implies PRE. Moreover, it has applications in achieving tight proofs for hash-based signatures (which was the reason for us to start looking into this). The paper will appear at ASIACRYPT 2019.