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Theories of working memory often disagree on the relationships between processing and storage, particularly on how heavily they rely on an attention-based limited resource. Some posit separation and specialization of resources resulting in minimal interference to memory when completing an ongoing processing task, while others argue for a greater overlap in the resources involved in concurrent tasks. Here, we present four experiments that investigated the presence or absence of dual-task costs for memory and processing. The experiments were carried in an adversarial collaboration in which researchers from three opposing theories collaboratively designed a set of experiments and provided differential predictions in line with each of their models. Participants performed delayed recall of aurally and visually presented letters and an arithmetic verification task either as single tasks or with the arithmetic verification task between presentation and recall of letter sequences. Single- and dual-task conditions were completed with and without concurrent articulatory suppression. A consistent pattern of dual-task and suppression costs was observed for memory, with smaller or null effects on processing. The observed data did not fit perfectly with any one framework, with each model having partial success in predicting data patterns. Implications for each of the models are discussed, with an aim for future research to investigate whether some combination of the models and their assumptions can provide a more comprehensive interpretation of the pattern of effects observed here and in relevant previous studies associated with each theoretical framework.