Compactness conjecture for heteroclinic orbits modulo time translation

Let $\mathcal{H}$ be the space of heteroclinic orbits of the semilinear parabolic equation, and let $Y_\lambda$ denote the ambient space in which these orbits are considered. Identify heteroclinic orbits that differ only by time translation. Compactness conjecture. The resulting space of heteroclinic orbits is compact in $Y_\lambda$. A compactness theorem of this kind is needed to develop a Floer homology for the space of heteroclinic orbits. The source notes that related compactness results are known for positive solutions centered on an equilibrium, while the general-sign case remains open.