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This paper uses convex integration with avoidance and transversalityarguments to prove the relative $h$-principle for closed$\mathrm{SL}(3;\mathbb{R})^2$ 3-forms on oriented 6-manifolds. As corollaries,it is proven that if an oriented 6-manifold $\mathrm{M}$ admits any$\mathrm{SL}(3;\mathbb{R})^2$ 3-form, then every degree 3 cohomology class on$\mathrm{M}$ can be represented by an $\mathrm{SL}(3;\mathbb{R})^2$ 3-form and,moreover, that the corresponding Hitchin functional on$\mathrm{SL}(3;\mathbb{R})^2$ 3-forms representing this class is necessarilyunbounded above. Essential to the proof of the $h$-principle is a carefulanalysis of the rank 3 distributions induced by an$\mathrm{SL}(3;\mathbb{R})^2$ 3-form and their interaction with generic pairsof hyperplanes. The proof also introduces a new property of sets in affinespace, termed macilence, as a method of verifying ampleness.

The relative $h$-principle for closed $\mathrm{SL}(3;\mathbb{R})^2$ 3-forms