A q-Clausen-type product identity for terminating basic hypergeometric sums

Let $n$ and $r$ be nonnegative integers with $r\leqslant n$, and let $q,x$ be parameters for which the displayed expressions are defined. The q-Clausen-type conjecture.

$$\begin{aligned} &\left(\sum_{k=r}^{n}\frac{(q^{-n};q)_k(x;q^2)_kq^k}{(q;q)_{k-r}(q;q)_{k+r}}\right)\left(\sum_{k=r}^{n}\frac{(q^{-n};q)_k(x;q^2)_kq^{(n+1)k-{k\choose2}}}{(q;q)_{k-r}(q;q)_{k+r}x^k}\right)\\\\ &=\frac{(-1)^r(q;q)_n^2(x;q^2)_rq^r}{(q;q)_{n-r}(q;q)_{n+r}(q^2/x;q^2)_rx^r}\sum_{k=r}^{n}\frac{(q^{-n};q)_k(q^{n+1};q)_k(x;q^2)_k(q^2/x;q^2)_kq^k}{(q;q)_{k-r}(q;q)_{k+r}(q;q)_{2k}}. \end{aligned}$$

The text calls this a numerical-experiment companion to a theorem; no proof or resolution is supplied.