Andrews–Bachraoui conjecture on the signed generating function for $a_2(n)$

Let $a_2”(n)$ be the signed count of the partitions described above, with each partition having $j$ parts greater than $1$ assigned weight $(-1)^{j+1}$. Define

$$\chi(n)=\begin{cases}-1&\text{if }n\equiv 0,1\pmod 4,\\1&\text{if }n\equiv 2,3\pmod 4.\end{cases}$$

Andrews–Bachraoui conjecture. There holds

$$\sum_{n\geq 0}a_2”(n)q^n=\frac{q}{1-q^2}+\frac{1}{1+q}\sum_{n\geq 0}\chi(n)q^{\binom{n+1}{2}+1}.$$

This conjecture was stated in the earlier paper cited in the source and concerns an unexpectedly simple expression for the signed generating function. No resolution is supplied in the given text.