  It was proposed in [(https://doi.org/10.1103/PhysRevLett.114.145301){Chen et
al., Phys. Rev. Lett. $\mathbf{114}$, 145301 (2015)}] that spin-2 chains
display an extended critical phase with enhanced SU$(3)$ symmetry. This
hypothesis is highly unexpected for a spin-2 system and, as we argue, would
imply an unconventional mechanism for symmetry emergence. Yet, the absence of
convenient critical points for renormalization group perturbative expansions,
allied with the usual difficulty in the convergence of numerical methods in
critical or small-gapped phases, renders the verification of this hypothetical
SU$(3)$-symmetric phase a non-trivial matter. By tracing parallels with the
well-understood phase diagram of spin-1 chains and searching for signatures
robust against finite-size effects, we draw criticism on the existence of this
phase. We perform non-Abelian density matrix renormalization group studies of
multipolar static correlation function, energy spectrum scaling, single-mode
approximation, and entanglement spectrum to shed light on the problem. We
determine that the hypothetical SU$(3)$ spin-2 phase is, in fact, dominated by
ferro-octupolar correlations and also observe a lack of Luttinger-liquid-like
behavior in correlation functions that suggests that is perhaps not critical.
We further construct an infinite family of spin-$S$ systems with similar
ferro-octupolar-dominated quasi-SU$(3)$-like phenomenology; curiously, we note
that the spin-3 version of the problem is located in a subspace of exact G$_2$
symmetry, making this a point of interest for search of Fibonacci topological
properties in magnetic systems.
