When I completed my first eligible Mindshifting May exercise it seems like I immediately got awarded the badge.
Looking at the code for the badge awarder it seems only the month is checked:
def award_to?(user)
user.solutions.published.joins(exercise: :track).
where('tracks.slug': TRACK_SLUGS).
where('
(MONTH(published_at) = 4 AND DAY(published_at) = 30) OR
(MONTH(published_at) = 5) OR
(MONTH(published_at) = 6 AND DAY(published_at) = 1)
').
where.not('exercises.slug': 'hello-world').
count >= 5
end
AFAIR I did a good chunk of the Pharo track a year (or two?) ago, so it would appear that these exercises actually were counted for the badge.