The mighty tuple II conjecture for quantified constraint satisfaction

Let $A$ be a finite domain, let $\Gamma$ be a constraint language on $A$, and let $\QCSP(\Gamma)$ denote the quantified constraint satisfaction problem for $\Gamma$. A mighty tuple II is the structural configuration defined in the paper, and a constraint language q-defines such a tuple when it can define it by quantified primitive-positive formulas. Mighty tuple II conjecture. $\QCSP(\Gamma)$ is PSpace-complete if and only if $\Gamma$ q-defines a mighty tuple II. The paper establishes related implications among mighty tuples and proves classifications in some restricted settings, but the converse implication from a mighty tuple I to a mighty tuple II is not known in general; this conjecture proposes the corresponding PSpace-completeness dichotomy.