Sectional-curvature spectral gap conjecture for Hodge Laplacians

Let $M^n$ be a closed Riemannian manifold satisfying

$$\mathrm{sec}\ge -1,\quad \mathrm{diam}\le d,\quad \mathrm{vol}\ge v.$$

Here $C_{P,k}$ denotes the Poincaré constant for $k$-forms. Sectional-curvature spectral gap conjecture. For every $k\ge 1$,

$$C_{P,k}\le C(n,d,k,v).$$

This conjecture seeks uniform bounds for all positive-degree Poincaré constants under sectional-curvature, diameter, and volume restrictions; the source presents it in its open-problems section and gives no resolution.