A weaker stable-set colorability conjecture for set systems

Let $r\geq 2$ and let $\mathcal{F}$ be a set system. Write $\mathcal{F}_{\widetilde{r-\operatorname{stab}}}$ for the family obtained by retaining the source's weakened notion of $r$-stability, let $\operatorname{KG}^r(\mathcal{F}_{\widetilde{r-\operatorname{stab}}})$ be the associated $r$-uniform Kneser hypergraph, let $\chi$ denote its chromatic number, and let $cd_r(\mathcal{F})$ denote the $r$-colorability defect of $\mathcal{F}$. The weaker conjecture. For every such $r$ and $\mathcal{F}$,

$$\chi\left(\operatorname{KG}^r\left(\mathcal{F}_{\widetilde{r-\operatorname{stab}}}\right)\right)\geq\left\lceil\frac{cd_r(\mathcal{F})}{r-1}\right\rceil.$$

The paper proposes this as a weaker version of the conjecture it disproves, but does not establish it; its status is therefore open.