The "mute conversation" button is only available in posts that are in your "Notifications" timeline. I should investigate:
exactly what it does. I am guessing the reason it only exists in the notifications timeline is that it only affects notifications (as in, it does not filter the conversation from your home timeline, it only filters it from your @s, which is still useful for people you don't follow replying to the thing)
if above is true, see about filtering from home and other timelines
either way, put the option in the three dot menu on replies to you in your home timeline (and the post focus view)
The "mute conversation" button is only available in posts that are in your "Notifications" timeline. I should investigate: