The minor inequality facet conjecture for circulant set covering polyhedra

Let $C_n^k$ be a circulant matrix, let $W\subset\mathbb Z_n$ define a relevant circulant minor isomorphic to $C_{n'}^{k'}$, and let $Q(C_n^k)$ denote the associated set covering polyhedron. The corresponding minor inequality is

$$\sum_{i\in W}2x_i+\sum_{i\notin W}x_i\geq\left\lceil\frac{n'}{k'}\right\rceil.$$

Here, $W$ is relevant when $n'\neq0\pmod {k'}$ and $\left\lceil n'/k'\right\rceil>\left\lceil n/k\right\rceil$. The minor inequality facet conjecture. A relevant minor inequality corresponding to a minor of $C_n^k$ isomorphic to $C_{n'}^{k'}$ defines a facet of $Q(C_n^k)$ if and only if $n'=1\pmod {k'}$. The conjecture seeks a complete characterization of when these non-boolean, non-rank inequalities are facet defining; the supplied source gives no evidence of resolution.