# Do Tukey and Games-Howell tests need additional p value corrections?

I was reading documentation of function posthocTGH (the "Note" section, where authors expressed doubt about additional corrections to p-values) and linked discussion (particularly this answer in which opinion that Games-Howell test does not need additional corrections) and this answer (that Conover-Iman test, which is a posthoc for Kruskal-Wallis tests, needs additional corrections) and got confused.

My question: Do (1) Tukey and (2) Games-Howell test (which are posthoc tests for ANOVA) need additional family-wise p-value (e.g., Holm, Bonferroni, etc.) corrections to prevent inflated I type error? Or is it sufficient to use p values of these tests with no additional corrections?

UPDATE: it seems that Tukey HSD test does not need additional corrections (link). Does the same apply to Games-Howell?