6-regular partition truncated-sum inequality

For any integer $n$, define

$$\alpha_n=\begin{cases}(-1)^m,&\text{if }n=3m(3m-1),\ m\in\mathbb Z,\\0,&\text{otherwise.}\end{cases}$$

Let $b_6(n)$ denote the $6$-regular partition function. 6-regular partition inequality. For $n\geqslant0$ and $k>0$,

$$(-1)^k\left(\alpha_n-\sum_{j=-(k-1)}^k(-1)^jb_6\bigl(n-j(3j-1)/2\bigr)\right)\geqslant0,$$

with strict inequality if $n\geqslant k(3k+1)/2$.

This is the combinatorial interpretation of a proposed coefficient-sign and nonvanishing property for a truncated pentagonal-number series associated with $6$-regular partitions; it remains open.