The regular-cardinal conjecture for sunflowerable dense linear orderings

Let $\kappa$ be a cardinal. A linear ordering is $\kappa$-dense when it has the density property at cardinality $\kappa$, and it is sunflowerable when it has the sunflower property.

The regular-cardinal conjecture for sunflowerable dense linear orderings. If $\kappa$ is regular, then every $\kappa$-dense linear ordering is sunflowerable.

The paper has a characterization of sunflowerable countable linear orderings, but explains that the uncountable case is not fully characterized; in particular, there are linear orderings that are $\mathbb{Q}_\kappa$-scattered but not $\kappa$-scattered. The stated implication is left open.