Exceptional negative coefficients conjecture for $D'(2,3,n)$

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 coefficient is negative when it is less than zero. The exceptional negative coefficients conjecture for $D'(2,3,n)$. The only negative coefficients of

$$\sum_{n\geq 0}D'(2,3,n)q^n$$

occur at $n=10$ and $n=22$. The claim refines the expected sign pattern for a series for which full positivity is not asserted.