On the complete bounds of $L_p$-Schur multipliers

We study the class $\mathcal{M}_p$ of Schur multipliers on the Schatten-von Neumann class $\mathcal{S}_p$ with $1 \leq p \leq \infty$ as well as the class of completely bounded Schur multipliers $\mathcal{M}_p^{cb}$. We first show that for $2 \leq p < q \leq \infty$ there exist $m \in \mathcal{M}_p^{cb}$ with $m \not \in \mathcal{M}_q$, so in particular the following inclusions that follow from interpolation are strict $\mathcal{M}_q \subsetneq \mathcal{M}_p$ and $\mathcal{M}^{cb}_q \subsetneq \mathcal{M}^{cb}_p$. In the remainder of the paper we collect computational evidence that for $p\not = 1,2, \infty$ we have $\mathcal{M}_p = \mathcal{M}_p^{cb}$, moreover with equality of bounds and complete bounds. This would suggest that a conjecture raised by Pisier is false.