The general vector-stability chromatic formula conjecture

Let $n,k$ be positive integers with $k\geq2$, and let $\vec{s}=(s_1,\ldots,s_k)$ be a positive integer vector with $s_i\geq2$ for $1\leq i\leq k-1$. Let $\mathrm{KG}^2(n,k)_{\vec{s}\textup{-stab}}$ denote the graph whose vertices are the $k$-subsets satisfying the vector-stability parameters $\vec{s}$. The vector-stability chromatic formula conjecture. If $n\geq\sum_{i=1}^k s_i$ and $s_k\leq2\cdot\min\{s_i:1\leq i\leq k-1\}$, then

$$\chi\left(\mathrm{KG}^2(n,k)_{\vec{s}\textup{-stab}}\right)=n-\left(\sum_{i=1}^{k-1}s_i\right)-\max\left\{0,s_k-\min\{s_i:1\leq i\leq k-1\}\right\}.$$

This conjecture is motivated by the paper's partial theorem and concerns a general formula for vector-stable Kneser graphs. The source states that a general formula remains elusive and gives no resolution of this conjecture.