The depth of Tsirelson's norm

Tsirelson’s norm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_T$ on $c_{00}$ is defined as the supremum over a certain collection of iteratively defined, monotone increasing norms $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_k$. For each positive integer $n$, the value $j(n)$ is the least integer $k$ such that for all $x\in {\mathbb{R}}^n$ (here ${\mathbb{R}}^n$ is considered as a subspace of $c_{00}$), $\mathrm{\parallel }x{\mathrm{\parallel }}_T=\mathrm{\parallel }x{\mathrm{\parallel }}_k$. In 1989 Casazza and Shura asked what is the order of magnitude of $j(n)$. It is known that $j(n)\in \mathcal{O}(\sqrt{n})$. We show that this bound is tight, that is, $j(n)\in \mathrm{\Omega }(\sqrt{n})$. Moreover, we compute the tight order of magnitude for some norms being modifications of the original Tsirelson’s norm.