The alternating inequality conjecture for $4$-regular partitions counted by $Q_2(n)$

Let $Q_2(n)$ denote the relevant counting function for $4$-regular partitions, and let $\chi_n$ denote the auxiliary sequence appearing in the conjecture. For integers $n,k\geqslant 0$, define the alternating finite sum

$$\sum_{j=0}^{2k-1}(-1)^{j(j+1)/2}Q_2\bigl(n-j(j+1)/2\bigr).$$

Alternating inequality conjecture. For $n,k\geqslant 0$, one has

$$(-1)^{k-1}\left(\sum_{j=0}^{2k-1}(-1)^{j(j+1)/2}Q_2\bigl(n-j(j+1)/2\bigr)-\chi_n\right)\geqslant 0,$$

with strict inequality if and only if $n\geqslant k(2k+1)$. Numerical evidence suggests that this gives a family of linear homogeneous inequalities analogous to earlier inequalities for partition functions; the conjecture remains unresolved in the supplied source.