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Current session-based recommender systems (SBRSs) mainly focus on maximizing recommendation accuracy, while few studies have been devoted to improve diversity beyond accuracy. Meanwhile, it is unclear how the accuracy-oriented SBRSs perform in terms of diversity. Besides, the asserted "trade-off" relationship between accuracy and diversity has been increasingly questioned in the literature. Towards the aforementioned issues, we conduct a holistic study to particularly examine the recommendation performance of representative SBRSs w.r.t. both accuracy and diversity, striving for better understanding the diversity-related issues for SBRSs and providing guidance on designing diversified SBRSs. Particularly, for a fair and thorough comparison, we deliberately select state-of-the-art non-neural, deep neural, and diversified SBRSs, by covering more scenarios with appropriate experimental setups, e.g., representative datasets, evaluation metrics, and hyper-parameter optimization technique. Our empirical results unveil that: 1) non-diversified methods can also obtain satisfying performance on diversity, which might even surpass diversified ones; and 2) the relationship between accuracy and diversity is quite complex. Besides the "trade-off" relationship, they might be positively correlated with each other, that is, having a same-trend (win-win or lose-lose) relationship, which varies across different methods and datasets. Additionally, we further identify three possible influential factors on diversity in SBRSs (i.e., granularity of item categorization, session diversity of datasets, and length of recommendation lists).