The sparsifying-transduction characterization of dense class properties

Let $\mathscr C$ be a class of TOWS graphs, and let $\mathsf{Sp}(\mathscr C)$ be the class obtained by the paper's sparsifying transduction. Let $\Pi_S$ be a downset of weakly sparse class properties: if $\mathscr C\in\Pi_S$ and $\mathscr D$ is a weakly sparse transduction of $\mathscr C$, then $\mathscr D\in\Pi_S$. Define the dense analogue $\Pi$ by requiring every weakly sparse transduction of $\mathscr C$ to belong to $\Pi_S$. Sparsifying-transduction conjecture. For every class $\mathscr C$ of TOWS graphs,

$$\mathscr C\in\Pi\qquad\iff\qquad\mathsf{Sp}(\mathscr C)\in\Pi_S.$$

This would transfer every downclosed weakly sparse class property to its dense analogue through $\mathsf{Sp}$; the supplied text gives no resolution status.