Stability of maximal relative projection constants

For positive integers $n\ge r$, let $\lambda (r,n)$ denote the \emph{maximal relative projection constant} of $r$-dimensional subspaces of ${\mathrm{\ell }}_{\mathrm{\infty }}^n$ and $\lambda (r)$ denote the \emph{maximal absolute projection constant}, respectively. It is known that for any fixed $r$, $\lambda (r,n)$ is a non-decreasing sequence with limit $\lambda (r)$ as $n\to \mathrm{\infty }$. A natural question is whether $\lambda (r,n)$ stabilizes at $\lambda (r)$ for some $n>r$. We prove that for any fixed $r$, $$\lambda (r,n)=\lambda (r)\text{for every}n\ge 2^r\left(\genfrac{}{}{0px}{}{r+1}{2}\right)\mathrm{.}$$ This answers a question of Basso. The technique used is of independent interest.