  Recent experiments [J. Guo et al., Phys. Rev. Lett.124,206602 (2020)] on
thermodynamic properties of the frustrated layered quantum magnet
SrCu$_2$(BO$_3$)$_2$ -- the Shastry-Sutherland material -- have provided strong
evidence for a low-temperature phase transition between plaquette-singlet and
antiferromagnetic order as a function of pressure. Further motivated by the
recently discovered unusual first-order quantum phase transition with an
apparent emergent O(4) symmetry of the antiferromagnetic and plaquette-singlet
order parameters in a two-dimensional "checkerboard J-Q" quantum spin model [B.
Zhao et al., Nat. Phys. 15, 678 (2019)], we here study the same model in the
presence of weak inter-layer couplings. Our focus is on the evolution of the
emergent symmetry as the system crosses over from two to three dimensions and
the phase transition extends from strictly zero temperature in two dimensions
up to finite temperature as expected in SrCu$_2$(BO$_3$)$_2$. Using quantum
Monte Carlo simulations, we map out the phase boundaries of the
plaquette-singlet and antiferromagnetic phases, with particular focus on the
triple point where these two order phases meet the paramagnetic phase for given
strength of the inter-layer coupling. All transitions are first-order in the
neighborhood of the triple points. We show that the emergent O(4) symmetry of
the coexistence state breaks down clearly when the interlayer coupling becomes
sufficiently large, but for a weak coupling, of the magnitude expected
experimentally, the enlarged symmetry can still be observed at the triple point
up to significant length scales. Thus, it is likely that the plaquette-singlet
to antiferromagnetic transition in SrCu$_2$(BO$_3$)$_2$ exhibits remnants of
emergent O(4) symmetry, which should be observable due to additional weakly
gapped Goldstone modes.
