Weaker $Q_2$ truncated-sum inequality

For any nonnegative integer $n$, define

$$\gamma_n=\begin{cases}\rho_m,&\text{if }n=m(m+1),\ m\in\mathbb N_0,\\0,&\text{otherwise.}\end{cases}$$

Let $Q_2(n)$ denote the partition function in the source. Weaker $Q_2$ truncated-sum inequality. For $n\geqslant0$ and $k>0$,

$$(-1)^{k-1}\left(\gamma_n-\sum_{j=-k}^k(-1)^jQ_2\bigl(n-j(3j-1)/2\bigr)\right)\geqslant0.$$

The source explicitly describes this as weaker than the preceding $Q_2$ conjecture; it remains open.