  Recently, Andrews, Chan, Kim and Osburn introduced the even strings and the
odd strings in the overpartitions. We show that their conjecture $A_k (n) \geq
B_k (n)$ holds for large enough positive integers n, where A_k(n) (resp.
B_k(n)) is the number of odd (resp. even) strings along the overpartitions of
n. We introduce m-strings and show that how this new combinatorial object is
related with another positivity conjecture of Andrews, Chan, Kim, and Osburn.
Finally, we confirm that the positivity conjecture is also true for large
enough integers.
