Does Grand Abolisher prevent opponents from activating abilities during your turn?

Grand Abolisher's ability reads: 'During your turn, your opponents can't cast spells or activate abilities of artifacts, creatures, or enchantments.' This is a static ability that creates a continuous restriction on what your opponents may do while it is your turn (CR 604.1).

The restriction covers activated abilities — those written in the form '[cost]: [effect]' (CR 602.1) — of permanents in the artifact, creature, or enchantment card types. This includes tap abilities, sacrifice abilities, and any other activated ability on those permanent types your opponents control.

Importantly, Grand Abolisher does not prevent opponents from activating abilities of lands or planeswalkers, nor does it stop triggered abilities from going on the stack, since triggered abilities are not 'activated' (CR 603.1). It also does not prevent opponents from activating mana abilities in cases where those mana abilities are on non-land permanents — actually, wait: it does prevent activation of mana abilities on creatures, artifacts, or enchantments, since the card text broadly covers all activated abilities of those types.

Concrete example: You control Grand Abolisher on your turn and attack. Your opponent controls a creature with a tap ability that deals damage (like a Prodigal Pyromancer). They cannot activate that tap ability during your turn at all — not even in response to your spells — because Grand Abolisher prohibits it.

Players attempting to work around Grand Abolisher should note that activated abilities on lands (e.g., cycling a land, or a utility land's ability) and planeswalker loyalty abilities are not blocked, since those card types are not listed in the restriction (CR 604.1, 602.1).