The tautologicality conjecture for effective classes of a logarithmic target

Let $\infty_\mathcal{X}$ be a logarithmic target with Gromov--Witten classes, and let $\infty$ be the associated target whose effective classes are under consideration. A class is tautological in cohomology (resp. Chow) when it lies in the corresponding tautological ring. Tautologicality conjecture. If $\infty_\mathcal{X}$ has tautological Gromov--Witten classes in cohomology (resp. Chow), then effective classes of $\infty$ are tautological in cohomology (resp. Chow). This would give a strong connection between tautological Gromov--Witten classes and effective classes; the supplied text does not indicate whether the assertion has been proved or disproved.