Quadratic growth of the stabilization threshold

Determine whether the minimum integer n_r such that lambda(r,n)=lambda(r) for every nge n_r satisfies n_r=O(r^2) as rtoinfty.

Background

For each positive integer r, the maximal relative projection constants lambda(r,n) form a non-decreasing sequence in n that converges to the maximal absolute projection constant lambda(r). The paper proves that stabilization occurs by n=N_r=2^rbinom{r+1}{2}, so the minimum stabilization threshold n_r satisfies n_rle 2^rbinom{r+1}{2}.

The authors note that obtaining a polynomial bound in r does not seem straightforward. The known exact stabilization thresholds for rin{1,2,3,7,23} are consistent with quadratic growth, motivating the unresolved question of whether n_r can always be bounded by a constant multiple of r^2.