Conjecture that all Weil–Petersson geodesics on Calabi–Yau threefold moduli are stable

Let the moduli space of a Calabi–Yau threefold be equipped with its Weil–Petersson geometry, and consider its geodesics. Stability conjecture. All Weil–Petersson geodesics on the moduli space of a Calabi–Yau threefold are stable. This conjecture is attributed in the source to the authors of the cited work and is presented in the context of special Kähler geometry and quantum gravity. The supplied text does not establish whether it has been proved or disproved.