Andrews' conjecture on a ${}_4\phi_3$ summation

Let $n\geq 0$, and let the basic hypergeometric series ${}_4\phi_3$ and the $q$-shifted factorials be defined by

$${}_4\phi_3\left[\begin{array}{c}a_1,a_2,a_3,a_4\\b_1,b_2,b_3\end{array};q,z\right]=\sum_{k=0}^{\infty}\frac{(a_1,a_2,a_3,a_4;q)_kz^k}{(q,b_1,b_2,b_3;q)_k},$$

where $(a_1,\ldots,a_m;q)_r=\prod_{i=1}^m\prod_{j=0}^{r-1}(1-a_iq^j)$. Andrews' conjecture. For $n\geq 0$, there holds

$${}_4\phi_3\left[\begin{array}{c}q^{-2n},\,a,\,b,\,q^{3-2n}/ab\\q^{2-2n}/a,\,q^{4-2n}/b,\,abq\end{array};q^2,q^2\right] =\frac{(a,-q;q)_n(b;q)_{n-1}(ab;q^2)_{n-1}\left(abq^{2n-2}(b-q^2)+abq^{n-1}(q-1)+q-b\right)}{q^{n+1}(1-abq^{2n-1})(ab;q)_{n-1}(a,b/q^2;q^2)_n}.$$

This summation is a basic hypergeometric identity conjectured by Andrews and is presented in the source as a conjecture; the supplied status is unknown, while this paper's abstract says that the conjecture is confirmed by the authors' proof.