Honda–Mondino conjecture on first Hodge-Laplacian eigenvalues

For given $m \in \mathop{\mathrm{{\Bbb N}}}\nolimits$, $\kappa \in \mathop{\mathrm{{\Bbb R}}}\nolimits$ and $v,D>0$, consider connected oriented closed Riemannian manifolds $(M^m,g)$ of dimension $m$ with Ricci curvature $\operatorname{Ric}_g\geq\kappa$, volume $\operatorname{vol}(M,g)\geq v$, and diameter $\operatorname{diam}(M,g)\leq D$. Honda–Mondino's conjecture. There exists a positive constant $C(m,\kappa,v,D)>0$, depending only on $m$, $\kappa$, $v$, and $D$, such that every such manifold satisfies

$$\lambda^{(1)}_1(M,g)\geq C(m,\kappa,v,D)>0.$$

The source presents this as a conjecture in the case $p=1$ after noting a positive lower bound under stronger assumptions in dimensions $m\leq4$; its resolution status is not specified in the supplied text.