The twin-width gap conjecture for random Latin square graphs

Let $G$ be a random Latin square graph, meaning the graph obtained from a randomly chosen $n\times n$ Latin square. Write $\operatorname{tww}(G)$ for its twin-width and $\operatorname{lb}_1(G)$ for its $\operatorname{lb}_1$-collapsibility parameter.

Twin-width gap conjecture. The twin-width of $G$ is asymptotically larger than $\operatorname{lb}_1(G)$, that is,

$$\operatorname{tww}(G)\gg \operatorname{lb}_1(G)$$

as $n$ tends to infinity.

This conjecture predicts that the upper bound obtained from $\operatorname{lb}_1$-collapsibility is not asymptotically tight for random Latin square graphs, and that twin-width and this collapsibility parameter can have substantially different asymptotic behavior. The supplied text gives no resolution, so the conjecture remains open.