Does a procedure controlling FDR at level $\alpha$ always reject at least as much as a procedure controlling FWE at $\alpha$? I understand the false discovery rate (FDR) is weakly less than the familywise error rate (FWE), and FDR is thus a less stringent way to control for type 1 errors.  However, will a procedure that ensures $FDR\leq \alpha$ necessarily reject any hypothesis rejected by a procedure that ensures $FWE\leq \alpha$? The Benjamini and Hochberg (1995) method rejects at least as much as Holm (1979) and Bonferroni methods, but it is unclear to me if this would be true for arbitrary methods controlling FDR and FWE.

Ok I found that the Benjamini–Yekutieli (2001) procedure, which controls FDR under arbitrary dependence of p values,  uses a lower cutoff than Benjamini and Hochberg (1995) and apparently lower than Holm, implying that some nulls rejected by Holm may get accepted by Benjamini–Yekutieli (2001).
