P-values high across clusters? [conceptual question]

[bhavyaac]

Hello! I have a conceptual question regarding the meaning of the p-values generated by CellphoneDB.

I understood from the protocols paper by Efremova et al. (2020) that the p-value indicates the likelihood of the interaction being specific to a given cluster / cell-type. Therefore, I would expect that low p-values would only be present between a few clusters for a given gene interaction. However, in my dot plot of CellphoneDB results, I obtain results such as these:

where interactions such as CD74_MIF and EGFR_AREG have low p-values across several combinations of clusters.

How is this possible? Please let me know what you think.