2 problems

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The Euler characteristic conjecture for elliptic complexes on $G_2$ - and $\Spin(7)$ -manifolds
Euler characteristic conjecture. The Euler number of $\mathcal{C}$ is
Conjectureopenelliptic-complexes g2-manifolds spin-7-manifoldsarXiv posted 2 days ago
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The existence conjecture for Ricci-flat Spin(7) metrics on generalized connected sums
Ricci-flat Spin(7) metric conjecture. There exists a Ricci-flat metric of holonomy $\operatorname{Spin}(7)$ on such manifolds.
Conjectureopenexceptional-holonomy ricci-flat-metrics spin-7-manifoldsarXiv posted 2 days ago

The Euler characteristic conjecture for elliptic complexes on G2G_2G2​- and \Spin(7)\Spin(7)\Spin(7)-manifolds