The cryptogenography problem, introduced by Brody, Jakobsen, Scheder, and Winkler (ITCS 2014), is to collaboratively leak a piece of information known to only one member of a group 1) without revealing who was the origin of this information and 2) without any private communication, neither during the process nor before. Despite several deep structural results, even the smallest case of leaking one bit of information present at one of two players is not well understood. Brody et al. gave a 2-round protocol enabling the two players to succeed with probability 1/3 and showed the hardness result that no protocol can give a success probability of more than 3/8. In this work, we show that neither bound is tight. Our new hardness result, obtained by a different application of the concavity method used also in the previous work, states that a success probability of better than 0.3672 is not possible. Using both theoretical and numerical approaches, we improve the lower bound to 0.3384, that is, give a protocol leading to this success probability. Unfortunately, already our smallest protocol beating the previous 1/3 success probability takes up 16 rounds of communication. The protocol leading to the bound of 0.3384 even in a compact representation consists of 18248 game states. These numbers suggest that the task of finding good protocols for the cryptogenography problem as well as understanding their structure is harder than what the simple problem formulation suggests.