What are methods for proving nonnegativity of q-hypergeometric functions? Specifically, I have a function of the type 4-phi-3, it is a terminating series: $$ {}_{4}\phi_3\left(\begin{matrix} q^{-i_1},q^{-j_1},zs_1^{-1}s_2 ,q z^{-1}s_1^{-1}s_2\\ s_2^{2},q^{1+j_2-j_1},s_1^{-2}q^{1-i_1-j_2}\end{matrix} \bigg|\, q,q\right). $$ I know from numerics that it should be nonnegative if $0<q<1, -1<s_1,s_2<0$, and $0\le z\le \min(s_1/s_2,s_2/s_1)$. Here $i_1,i_2,j_1,j_2$ are nonnegative integers with condition $i_1+j_2=j_1+i_2$.

However, in the expansion of the function the individual terms are not all nonnegative. They seem to be of alternating signs, and are not monotone either. I tried various Watson's transformations turning the function into 8-phi-7, but so far did not succeed. Any help appreciated.