Equidistant-number asymptotic limit

Determine whether the limit describing the asymptotic maximum gap between \(k\)-independence and \(k\)-equidistant numbers equals one for every \(k>1\).

Background

The survey defines $AE(n,k)$ as the maximum difference between the $(k-1)$-independence number and the $k$-equidistant number among graphs of order $n$. The relevant normalized limit is known to equal one for $k=1$, but its value for larger $k$ is unresolved.