Eventual positivity conjecture for $D'(k,m,n)$

Let $D'(k,m,n)$ be defined by

$$\sum_{n\geq 0}D'(k,m,n)q^n=q^{-m}\sum_{n\geq 0}C'(k,m,n)q^n-\frac{(q^2,q^{2k};q^2)_\infty}{(q;q^2)_\infty^2},$$

where $k$ and $m$ are positive integers. A series is eventually positive when all coefficients are nonnegative from some index onward. The eventual positivity conjecture for $D'(k,m,n)$. If $k>m$, then

$$\sum_{n\geq 0}D'(k,m,n)q^n$$

is eventually positive. The paper gives no proof of this general assertion, leaving it as an open conjecture.