The lunch box that Ruth found was placed inside a freezer bag, which is likely the same one that contained Haruki #1’s. Haruki #1’s old watch is a symbol of passing time, and by asking Nao to promise to keep it wound and running, Jiko wanted her to always remember that time was rushing by. Given that Ruth is the one in possession of the watch now, she’ll likely take this reminder to heart as she reads it in Nao’s diary.