This paper presents a case in conflict management where a neutral, benevolent mediator should propose a biased peaceful split between two potential adversaries despite that the two are stochastically identical and that the mediator puts equal weight on their welfare.  A biased proposal, when rejected, induces asymmetric posterior beliefs conducive to mitigating the detriment of the conflict, modeled as an all-pay auction.  This positive effect of induced asymmetry, however, may be offset by the negative effect on the probability of peace settlements, as the paper notes two classes of biased proposals inferior to the unbiased one.  We provide an explicit biased proposal that outperforms the unbiased one.  This better alternative is so lopsided that the favored player always accepts it, while only the weak type of the opponent may accept it.