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We do this at the moment by splitting up the company in two
groups.
- managers
- oters

People in (Group) Managers have the privelage to be able to
close

Of cource your proposed way is better, because you still
have a better overview for the whole company.

Thsi can be bypassed imho by using this two group thing and
assigning an additional function "close" which is in fact
a "Wait". The final close should than be set to automatic and
this will than be done by the managers.

This way there is still a rather complete list left and the time
can be set to a week or 2.