Quantum Knizhnik–Zamolodchikov equation, totally symmetric self-complementary plane partitions, and alternating sign matrices

We present multiple-residue integral formulas for partial sums in the basis of link patterns of the polynomial solution of the level-$1$ $U_q(\widehat{\mathfrak{sl}_2})$ quantum Knizhnik–Zamolodchikov equation at arbitrary values of the quantum parameter $q$. These formulas allow rewriting and generalizing a recent conjecture of Di Francesco connecting these sums to generating polynomials for weighted totally symmetric self-complementary plane partitions. We reduce the corresponding conjectures to a single integral identity, yet to be proved.