Dependent internal tree-ordering expansions and bounded twin-width

Let $\mathscr C$ be a hereditary weakly sparse class of graphs. An internal tree-ordering expansion of a graph is an expansion by the forest-ordering defined by a spanning forest; $\mathscr C$ has a dependent internal tree-ordering expansion when such expansions form a monadically dependent class. Internal tree-ordering conjecture. $\mathscr C$ has a dependent internal tree-ordering expansion if and only if $\mathscr C$ has bounded twin-width. This would characterize bounded twin-width for hereditary weakly sparse graph classes through dependent internal tree-orderings; the supplied text gives no resolution status.