  Higgs plus multi-gluon amplitudes in QCD can be computed in an effective
Lagrangian description. In the infinite top-mass limit, an amplitude with a
Higgs and $n$ gluons is computed by the form factor of the operator ${\rm Tr}\,
F^2$. Up to two loops and for three gluons, its maximally transcendental part
is captured entirely by the form factor of the protected stress tensor
multiplet operator $\mathcal{T}_2$ in $\mathcal{N}=4$ supersymmetric Yang-Mills
theory. The next order correction involves the calculation of the form factor
of the higher-dimensional, trilinear operator ${\rm Tr}\, F^3$. We present
explicit results at two loops for three gluons, including the subleading
transcendental terms derived from a particular descendant of the Konishi
operator that contains ${\rm Tr}\, F^3$. These are expressed in terms of a few
universal building blocks already identified in earlier calculations. We show
that the maximally transcendental part of this quantity, computed in
non-supersymmetric Yang-Mills theory, is identical to the form factor of
another protected operator, $\mathcal{T}_3$, in the maximally supersymmetric
theory. Our results suggest that the maximally transcendental part of Higgs
amplitudes in QCD can be entirely computed through $\mathcal{N}=4$ super
Yang-Mills.
