  We look into the quantum phase diagram of a spin-$\frac{1}{2}$
antiferromagnet on the square lattice with degenerate Shastry-Sutherland ground
states, for which only a schematic phase diagram is known so far. Many exotic
phases were proposed in the schematic phase diagram by the use of exact
diagonalization on very small system sizes. In our present work, an important
extension of this antiferromagnet is introduced and investigated in the
thermodynamic limit using triplon mean-field theory. Remarkably, this
antiferromagnet shows a stable plaquette spin-gapped phase like the original
Shastry-Sutherland antiferromagnet, although both of these antiferromagnets
differ in the Hamiltonian construction and ground state degeneracy. We propose
a sublattice columnar dimer phase which is stabilized by the second and third
neighbor antiferromagnetic Heisenberg exchange interactions. There are also
some commensurate and incommensurate magnetically ordered phases, and other
spin-gapped phases which find their places in the quantum phase diagram.
Mean-field results suggest that there is always a level-crossing phase
transition between two spin gapped phases, whereas in other situations, either
a level-crossing or a continuous phase transition happens.
