Tautological Chow conjecture for stable maps to general flag varieties

Let $G$ be a semisimple algebraic group, let $P$ be a parabolic subgroup, and let $X=G/P$ be a general flag variety. For nonnegative integers $n$ and curve class $\beta$, write $\mathcal{\overline M}_{0,n}(X,\beta)$ for the moduli space of stable maps to $X$. A Chow class is tautological if it belongs to the tautological ring generated by the standard tautological classes and boundary classes. Tautological Chow conjecture. All rational Chow classes of $\mathcal{\overline M}_{0,n}(X,\beta)$ are tautological. The paper explains that the analogous result is proved for the flag varieties considered there and expects it to hold for general flag varieties over any field; the conjecture remains unresolved in the supplied text.