Proof of the Andrews-El Bachraoui positivity conjecture

We prove that for $k\ge 1$, all coefficients in the expansion of the series $$\sum _{n\ge 0}\frac{(q^{2n+2},q^{2n+2k};q^2{)}_{\mathrm{\infty }}}{(q^{2n+1};q^2{)}_{\mathrm{\infty }}^2}{q}^{2n}$$ are positive, by $q$-hypergeometric means. This confirms a recent conjecture of Andrews and El Bachraoui.