Siggers-behavior characterization of k-consistency for network satisfaction problems

Let ${\bf A}$ be a finite symmetric relation algebra with a normal representation. Say that ${\bf A}$ admits a Siggers behavior when it satisfies the Siggers condition described in the paper; the condition that is being weakened is the existence of all $1$-cycles in ${\bf A}$. Let $\operatorname{NSP}({\bf A})$ denote the network satisfaction problem for ${\bf A}$.

Siggers-behavior conjecture. Dropping the requirement that ${\bf A}$ has all $1$-cycles should yield a necessary and sufficient condition for solvability of $\operatorname{NSP}({\bf A})$ by the $k$-consistency procedure; explicitly, ${\bf A}$ admits a Siggers behavior if and only if there is some $k\in\mathbb{N}$ such that $\operatorname{NSP}({\bf A})$ can be solved by the $k$-consistency procedure.

The conjecture seeks a characterization of when bounded-width consistency methods solve network satisfaction problems for finite symmetric relation algebras with normal representations. The paper establishes a sufficient condition involving both all $1$-cycles and a Siggers behavior, while the necessity and sufficiency after dropping the $1$-cycle condition remain open.