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We investigate the physical properties of $G_2$-symmetric integrable chainswith local degrees of freedom in the fundamental representation; given thetypical connection between integrability and critical points, we test themodel's properties against a hypothesis of conformal-invariant long-distancebehavior. Leveraging an embedding between the $G_2$ exceptional Lie algebra and$SU(2)$-symmetric chains with local spin-3 representations, we performnumerical analyses via exact diagonalization (ED) targeted at specific spinsectors, as well as via non-Abelian density-matrix renormalization group(DMRG). A basic study of the momentum-resolved ED spectrum suggests thelow-energy system is effectively described by a $(G_2)_1$ Wess--Zumino--Witten(WZW) theory, but we find challenges in further numerical characterization ofconformal data. The study and control of the phenomenology of this model mayhave implications for the development of accessible models for Fibonaccianyons.

$G_2$ Integrable Point Characterization via Isotropic Spin-3 Chains