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.
