Acyclicity conjecture for graphs of affine reversal sets

Let $w\in \widetilde{\mathfrak{S}}_n$, let $\mathcal{C}_w(n,k)$ denote the relevant collection of consistent subsets, and let $R\in \mathcal{C}_w(n,k)$. Define $G_R$ by including edges $X\to Y$ whenever $X<_{\mathcal{P}_w(n,k)}Y$ is a congruence relation. Acyclicity conjecture. For any $R\in \mathcal{C}_w(n,k)$, the graph $G_R$ is acyclic. This is the obstruction identified in the source to proving a complete affine analogue of the corresponding nonlongest-permutation result; the source reports computational verification for $1\leq k\leq 8$ and affine permutations of bounded lengths, but gives no general proof.