  We introduce a quantum spin-1/2 model with many-body correlated
Heisenberg-type interactions on the 2D square lattice, designed to host a
plaquette valence-bond solid (PVBS) ground state breaking $\mathbb{Z}_4$
symmetry. We carry out a detailed quantum Monte Carlo study of the quantum
phase transition between the antiferromagnetic (AFM) and PVBS states. We find a
first-order transition, in contrast to previously studied continuous
'deconfined' transitions between the AFM and columnar valence-bond solids. We
show that the coexistence state at the AFM--PVBS transition is unexpectedly
associated with SO(5) symmetry, which may indicate that the transition is
connected to a deconfined critical point. We also discuss the first-order
transition in the context of a recent proposal of spinons with fracton
properties in the PVBS state, concluding that the fracton scenario is unlikely.
Further, we discover a novel type of eight-fold degenerate VBS phase that
breaks the remaining $\mathbb{Z}_2$ symmetry of the PVBS phase. The PVBS phase
can then be regarded as an intermediate 'vestigial' phase, and we develop a
general graph-theoretic approach to describe the two-stage discrete symmetry
breaking. At finite temperature, we observe fluctuation-induced first-order
transitions, which are hallmarks of vestigial phase transitions. We also
mention possible connections to the SO(5) theory of high-T$_{\rm c}$
superconductivity.
