Recently, Andrews, Chan, Kim and Osburn introduced the even strings and the odd strings in the overpartitions. We show that their conjecture \[ A_k (n) \ge B_k (n) \] 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 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.