6-regular partition inequalities involving $\beta_n$

For any integer $n$, define

$$\beta_n=\begin{cases}(-1)^m,&\text{if }n=m(3m-2),\ m\in\mathbb Z,\\0,&\text{otherwise.}\end{cases}$$

Let $b_6(n)$ denote the $6$-regular partition function, and let $(\rho_j)_{j\geqslant0}$ be the coefficient sequence in the source. 6-regular partition inequalities. For $n,k\geqslant0$,

$$\sum_{j=0}^{3k}\rho_jb_6\bigl(n-j(j+1)/2\bigr)\geqslant\beta_n,$$

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

$$\sum_{j=0}^{3k+2}\rho_jb_6\bigl(n-j(j+1)/2\bigr)\leqslant\beta_n,$$

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

These inequalities are proposed as a combinatorial interpretation of coefficient-sign and nonvanishing conjectures for a truncated $6$-regular partition series; they remain open.