The exponential deficit conjecture for near-optimal Kneser graph colorings

Let $KG_{n,k}$ be the Kneser graph on the $k$-element subsets of $[n]$, with edges joining disjoint sets. Consider a subset of its vertices that is properly colorable with $n-2k+1$ colors. Exponential deficit conjecture. The largest possible size of such a subset is at most

$$\binom{n}{k}-c^k,$$

where $c>1$ is an absolute constant. This would strengthen the Lovász lower bound $\chi(KG_{n,k})\geqslant n-2k+2$ by asserting that deleting exponentially many vertices is necessary before an $(n-2k+1)$-coloring becomes possible. The corresponding edge version is posed immediately beforehand as a separate minimization problem.