Stabilization of Tsirelson-type norms on ${\mathrm{\ell }}_p$ spaces

We consider classical Tsirelson-type norms of $T[A_r,\theta ]$ and their modified versions on ${\mathrm{\ell }}_p$ spaces. We show that for any $1<p<\mathrm{\infty }$ there is a constant ${\lambda }_p$ such that considered Tsirelson-type norms do not ${\lambda }_p$-distort any of subspaces of ${\mathrm{\ell }}_p$.