The prevalence of sexual reproduction remains one of the most perplexing phenomena in evolutionary biology. The deterministic mutation hypothesis postulates that sexual reproduction will be advantageous under synergistic epistasis, a condition in which mutations cause a greater reduction in fitness when combined than would be expected from their individual effects. The inverse condition, antagonistic epistasis, correspondingly is predicted to favor asexual reproduction. To assess this hypothesis, we introduce a finite population evolutionary process that combines a recombination modifier formalism with a gene-regulatory network model. We demonstrate that when reproductive mode and epistasis are allowed to coevolve, asexual reproduction outcompetes sexual reproduction. In addition, no correlation is found between the level of synergistic epistasis and the fixation time of the asexual mode. However, a significant correlation is found between the level of antagonistic epistasis and asexual mode fixation time. This asymmetry can be explained by the greater reduction in fitness imposed by sexual reproduction as compared with asexual reproduction. Our findings present evidence and suggest plausible explanations that challenge both the deterministic mutation hypothesis and recent arguments asserting the importance of emergent synergistic epistasis in the maintenance of sexual reproduction.