Optimal square size for separating the operator and completely bounded norms of Schur multipliers on $S^4$

We determine the smallest square size at which the operator norm and the completely bounded norm of a Schur multiplier on $S^4$ can differ. More precisely, for $A=\begin{bmatrix}1&2&0\\1&-2&0\\2&2\mathrm{i}&0\end{bmatrix}$ we prove \[ \|M_A\|_{S^4_3\to S^4_3}^2\leq\frac{1119}{250} <\left(\frac{18983