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This paper combines explicit local calculations with covering arguments toprove the unboundedness above and below (in a logarithmic sense) of theDonaldson-Hitchin functionals on $\mathrm{G}_2$ 4-forms,$\widetilde{\mathrm{G}}_2$ 3-forms and $\widetilde{\mathrm{G}}_2$ 4-forms, overcompact manifolds (or, more generally, orbifolds) with boundary. In addition,the Donaldson-Hitchin functional on $\mathrm{G}_2$ 3-forms over compactmanifolds (or orbifolds) with boundary is shown to be unbounded below. Asscholia, the critical points of the functionals on $\mathrm{G}_2$ 4-forms,$\widetilde{\mathrm{G}}_2$ 3-forms and $\widetilde{\mathrm{G}}_2$ 4-forms areshown to be saddles, and initial conditions of the Laplacian coflow which donot lead to convergent solutions are shown to be dense.

Unboundedness above and below of the Donaldson-Hitchin functionals on $\mathrm{G}_2$- and $\widetilde{\mathrm{G}}_2$-forms