  This paper combines explicit local calculations with covering arguments to
prove the unboundedness above and below (in a logarithmic sense) of the
Donaldson-Hitchin functionals on $\mathrm{G}_2$ 4-forms,
$\widetilde{\mathrm{G}}_2$ 3-forms and $\widetilde{\mathrm{G}}_2$ 4-forms, over
compact manifolds (or, more generally, orbifolds) with boundary. In addition,
the Donaldson-Hitchin functional on $\mathrm{G}_2$ 3-forms over compact
manifolds (or orbifolds) with boundary is shown to be unbounded below. As
scholia, 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 are
shown to be saddles, and initial conditions of the Laplacian coflow which do
not lead to convergent solutions are shown to be dense.
