Lott's conjecture on lower bounds for Hodge-Laplacian eigenvalues

Let $n\in\mathbb{N}$, $\kappa\in\mathbb{R}$, and $\nu,D>0$. For a connected oriented closed Riemannian manifold $(M^n,g)$, let $\mu^{(p)}_1$ denote the first nonzero eigenvalue of the Hodge-Laplacian on $p$-forms. Assume

$$\operatorname{sec}_g\geq\kappa,\qquad \operatorname{diam}(M)\leq D,\qquad \operatorname{Vol}(g)\geq\nu.$$

Lott's conjecture. There exists a positive constant $C(n,\kappa,\nu,D)>0$, depending only on $n$, $\kappa$, $\nu$, and $D$, such that

$$\mu^{(p)}_1\geq C(n,\kappa,\nu,D),\qquad \text{for all }p=0,1,\ldots,n.$$

This conjecture asks whether a lower sectional-curvature bound, together with diameter and volume bounds, suffices to give uniform positive lower bounds for the first nonzero Hodge-Laplacian eigenvalues. It is an alternative to the known result under a two-sided sectional-curvature bound, and remains open as far as the authors know.