  We continue recent efforts to discover examples of deconfined quantum
criticality in one-dimensional models. In this work we investigate the
transition between a $\mathbb{Z}_3$ ferromagnet and a phase with valence bond
solid (VBS) order in a spin chain with $\mathbb{Z}_3\times\mathbb{Z}_3$ global
symmetry. We study a model with alternating projective representations on the
sites of the two sublattices, allowing the Hamiltonian to connect to an exactly
solvable point having VBS order with the character of SU(3)-invariant singlets.
Such a model does not admit a Lieb-Schultz-Mattis theorem typical of systems
realizing deconfined critical points. Nevertheless, we find evidence for a
direct transition from the VBS phase to a $\mathbb{Z}_3$ ferromagnet.
Finite-entanglement scaling data are consistent with a second-order or weakly
first-order transition. We find in our parameter space an integrable lattice
model apparently describing the phase transition, with a very long, finite,
correlation length of 190878 lattice spacings. Based on exact results for this
model, we propose that the transition is extremely weakly first order, and is
part of a family of DQCP described by walking of renormalization group flows.
