  Let $\mathbf{E}_n: \mathcal{M} \to \mathcal{M}_n$ and $\mathbf{E}_m:
\mathcal{N} \to \mathcal{N}_m$ be two sequences of conditional expectations on
finite von Neumann algebras. The optimal weak Orlicz type of the associated
strong maximal operator $\mathcal{E} = (\mathbf{E}_n\otimes
\mathbf{E}_m)_{n,m}$ is not yet known. In a recent work of Jose Conde and the
two first-named authors, it was show that $\mathcal{E}$ has weak type $(\Phi,
\Phi)$ for a family of functions including $\Phi(t) = t \, \log^{2+\varepsilon}
t$, for every $\varepsilon > 0$. In this article, we prove that the weak Orlicz
type of $\mathcal{E}$ cannot be lowered below $L \log^2 L$, meaning that if
$\mathcal{E}$ is of weak type $(\Phi, \Phi)$, then $\Phi(s) \not\in o(s \,
\log^2 s)$. Our proof is based on interpolation. Namely, we use recent
techniques of Cadilhac/Ricard to formulate a Marcinkiewicz type theorem for
maximal weak Orlicz types. Then, we show that a weak Orlicz type lower than $L
\log^2 L$ would imply a $p$-operator constant for $\mathcal{E}$ smaller than
the known optimum as $p \to 1^{+}$.
