Identifying signals that replicate across multiple studies is a fundamental challenge in modern data analysis. Partial conjunction (PC) testing addresses this problem by assessing whether a signal is present in at least a specified number of studies. The AdaFilter framework of Wang et al. (2022, Annals of Statistics 50, 1890-1909) improves the power of PC testing through adaptive filtering, but its family-wise error rate (FWER)-controlling procedures rely on conservative Bonferroni corrections at two stages: (i) the construction of PC $p$-values and filtering statistics, and (ii) the multiplicity adjustment used to control the FWER. We show that both stages can be improved using the Šidák correction. Under the same independence assumptions as the original AdaFilter framework, our improvements preserve finite-sample FWER control and guarantee at least as many rejections as their Bonferroni counterparts for every realisation of the data. Simulations and a real data analysis demonstrate the practical benefits of these improvements.

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