The probabilistic universal graph conjecture for stable matrix families

Let $\mathcal M$ be a sequence of $n \times n$ Boolean matrices. For a sequence of matrices, let $\operatorname{cl}(\mathcal M)$ be the set of all square submatrices of matrices in the sequence, and let $\operatorname{cl}(\mathcal M)_n$ denote its $n \times n$ members. Let $\operatorname{R}(\mathcal M)$ denote the randomized communication complexity of the sequence. If $\mathrm{GT}_k$ is the $k \times k$ Greater-Than matrix defined by $\mathrm{GT}_k(i,j)=1$ if and only if $i \leq j$, then $\mathcal M$ is stable when $\mathrm{GT}_k \notin \operatorname{cl}(\mathcal M)$ for some $k \in \mathbb N$. Probabilistic Universal Graph Conjecture. One has

$$\operatorname{R}(\mathcal M)=O(1)$$

if and only if $\mathcal M$ is stable and

$$|\operatorname{cl}(\mathcal M)_n|\leq 2^{O(n \log n)}.$$

The statement is the communication-complexity formulation of the probabilistic version of the Implicit Graph Conjecture proposed by Harms, Wild, and Zamaraev. The paper's abstract states that this conjecture is refuted, so the claimed equivalence is not valid in general.