The almost-stable Kneser hypergraph colorability bound

Let $[n]=\{1,\ldots,n\}$, let $\mathcal G$ be a hypergraph over the ground set $[n]$, and let $r\geq s\geq 2$. Write $\mathcal G_{\widetilde{\{s\text{-stable}\}}}$ for the corresponding almost-stable restriction, let $\operatorname{KG}^r(\mathcal G_{\widetilde{\{s\text{-stable}\}}})$ be its $r$-uniform Kneser hypergraph, let $\chi$ denote chromatic number, and let $\operatorname{cd}^r(\mathcal G)$ denote the $r$-colorability defect. The almost-stable Kneser hypergraph conjecture. Every such hypergraph satisfies

$$\chi\left(\operatorname{KG}^r\left(\mathcal G_{\widetilde{\{s\text{-stable}\}}}\right)\right)\geq\left\lceil\frac{\operatorname{cd}^r(\mathcal G)}{r-1}\right\rceil.$$

This conjecture is proposed in the paper as a partial answer to a problem concerning how large the gap between the colorability-defect bound and the chromatic number can be. The supplied text gives no resolution.