Stability of maximal relative projection constants

For positive integers $n\ge r$, let $λ(r,n)$ denote the \emph{maximal relative projection constant} of $r$-dimensional subspaces of $\ell_\infty^n$ and $λ(r)$ denote the \emph{maximal absolute projection constant}, respectively. It is known that for any fixed $r$, $λ(r,n)$ is a non-decreasing sequen