Associative memories are data structures that allow retrieval of previously stored messages given part of their content. They thus behave similarly to human brain's memory that is capable for instance of retrieving the end of a song given its beginning. Among different families of associative memories, sparse ones are known to provide the best efficiency (ratio of the number of bits stored to that of bits used). Nevertheless, it is well known that non-uniformity of the stored messages can lead to dramatic decrease in performance. Recently, a new family of sparse associative memories achieving almost-optimal efficiency has been proposed. Their structure induces a direct mapping between input messages and stored patterns. In this work, we show the impact of non-uniformity on the performance of this recent model and we exploit the structure of the model to introduce several strategies to allow for efficient storage of non-uniform messages. We analyse and discuss the methods introduced and we envision future work in this context.