Determine whether the finite matrix-level bound is optimal

Determine whether the matrix level $N_m=\left\lceil\frac{2m}{3}\right\rceil$ in the finite-level criterion for complete contractivity of projections on Haagerup noncommutative $L^{2m}$-spaces is optimal.

Background

For an even exponent $p=2m$ with $m\geq2$, the paper proves that $N_m=\lceil 2m/3\rceil$-contractivity implies complete contractivity for projections on Haagerup noncommutative L^{2m}-spaces over arbitrary von Neumann algebras. The argument supplies a sufficient finite matrix level by extracting alternating products through a Fourier-coefficient construction.

The paper explicitly does not establish that this level is minimal. In particular, the known Schatten-space result shows that level 2 suffices for every $1<p<\infty$, $p\neq2$, suggesting that the bound obtained here may not be optimal in broader settings.