Euler partition inequalities involving $Q_2(n)$

For $n,k\geqslant0$, let $p(n)$ denote Euler's partition function, let $Q_2(n)$ denote the partition function used in the source, and let $(\rho_j)_{j\geqslant0}$ be the coefficient sequence appearing in the preceding truncated-series definition. Euler partition inequalities.

$$\sum_{j=0}^{3k}\rho_jp\bigl(n-j(j+1)\bigr)\geqslant Q_2(n),$$

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

$$\sum_{j=0}^{3k+1}\rho_jp\bigl(n-j(j+1)\bigr)\leqslant Q_2(n),$$

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

These inequalities are presented as a combinatorial interpretation of the preceding coefficient conjectures and would provide infinite families of linear inequalities for partition functions; they remain open.