  Recently, Andrews, Chan, Kim and Osburn introduced a $q$-series $h(q)$ for
the study of the first positive rank and crank moments for overpartitions. They
conjectured that for all integers $m \geq 3$, \begin{equation*}\label{hqcon}
\frac{1}{(q)_{\infty}} (h(q) - m h(q^{m})) \end{equation*} has positive power
series coefficients for all powers of $q$. Byungchan Kim, Eunmi Kim and
Jeehyeon Seo provided a combinatorial interpretation and proved it is
asymptotically true by circle method. In this note, we show this conjecture is
true if $m$ is any positive power of $2$, and we show that in order to prove
this conjecture, it is only to prove it for all primes $m$. Moreover we give a
stronger conjecture. Our method is very simple and completely different from
that of Kim et al.
