Lott's lower-bound conjecture for 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 sectional curvature $K_g\geq\kappa$, volume $\operatorname{vol}(M,g)\geq v$, and diameter $\operatorname{diam}(M,g)\leq D$. Lott'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

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

for all $p=0,1,\dots,m$. This is an open problem for sectional curvature bounded below; the paper notes that it would follow from the still-unpublished Lipschitz stability theorem for Alexandrov spaces, while existing results establish only more restricted cases.