# Ricci curvature of the Weil-Petersson metric?

Let $$\omega_{\text{WP}}$$ denote the Weil-Petersson metric associated to a family of Calabi-Yau manifolds. That is, let $$f : X \to Y$$ be a surjective holomorphic map with connected fibres such that, over the regular locus, the fibres $$X_y$$ of $$f^{\circ} : X^{\circ} \to Y^{\circ}$$ are smooth compact Kähler manifolds with $$c_1 =0$$ in the $$H^2_{\text{DR}}(X_y, \mathbb{R})$$.

Let $$D$$ be the associated classifying space, which we know is a symmetric space of non-compact type. Let $$\omega_H$$ denote the invariant metric on $$D$$, which is referred to as the Hodge metric. Lu-Sun showed that the Ricci curvature of $$\omega_{\text{WP}}$$ is pinched according to: $$- \omega_H \leq \text{Ric}(\omega_{\text{WP}} ) \leq \omega_H,$$ and $$2\omega_{\text{WP}} \leq \omega_H$$.

Question: Is $$\text{Ric}(\omega_{\text{WP}})$$ honestly bounded from below, i.e., does there exist a constant $$C>0$$ such that $$\text{Ric}(\omega_{\text{WP}}) \geq - C \omega_{\text{WP}}?$$

I'd be interested in any results concerning the Ricci curvature of the Weil-Petersson metric, even for non-Calabi-Yau families.