Chromatic lower-bound conjecture for almost stable general Kneser hypergraphs

Let $n\ge 1$, $r\ge 2$, and $s\ge r$, and let $\cal F$ be a family of subsets of $[n]$. For the almost $s$-stable subfamily $\cal F_s$, let $\operatorname{KG}^r(\cal F_s)$ be the $r$-uniform Kneser hypergraph and let $\operatorname{ecd}^s(\cal F)$ denote the equitable $r$-colorability defect of $\cal F$. The chromatic lower-bound conjecture. One has

$$\chi\bigl(\operatorname{KG}^r(\cal F_s)\bigr)\ge \left\lceil \frac{\operatorname{ecd}^s(\cal F)}{r-1}\right\rceil.$$

The paper proves this bound when $r$ is a power of $2$ and $s$ is a multiple of $r$; the conjecture proposes the same inequality for all $s\ge r$.