Asymptotic conjecture for s-distance sets in Johnson spaces

Let $\mathcal{J}^{n,w}$ be the Johnson space of $w$-subsets of an $n$-element set, and let $A(\mathcal{J}^{n,w},s)$ denote the maximum size of an $s$-distance set in this space. Johnson-space asymptotic conjecture. For all integers $w>s>0$ and sufficiently large $n$, with the threshold allowed to depend on $w$ and $s$,

$$A(\mathcal{J}^{n,w},s)=\binom{n-w+s}{s}.$$

This conjecture proposes the eventual exact value of the largest $s$-distance sets in Johnson spaces. The source presents it as a general conjecture based on the paper's exact results and known bounds; no resolution is given.