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Laboratory data from conflict tasks, e.g. Simon and Eriksen tasks, reveal differences in response time distributions under different experimental conditions. Only recently have evidence accumulation models successfully reproduced these results, in particular the challenging delta plots with negative slopes. They accomplish this with explicit temporal dependencies in their structure or activation functions. In this work, we introduce an alternative approach to the modeling of decision-making in conflict tasks exclusively based on inhibitory dynamics within a dual-route architecture. We consider simultaneous automatic and controlled drift diffusion processes, with the latter inhibiting the former. Our proposed Dual-Route Evidence Accumulation Model (DREAM) achieves equivalent performance to previous works in fitting experimental response time distributions despite having no time-dependent functions. The model can reproduce conditional accuracy functions and delta plots with positive and negative slopes. The implications of these results, including an interpretation of the parameters and potential links to perceptual representations, are discussed. We provide Python code to fit DREAM to experimental data. [ABSTRACT FROM AUTHOR]