A simple spin-12 frustrated antiferromagnetic Heisenberg model (AFHM) on the square lattice—including chiral plaquette cyclic terms—was argued to host a bosonic Kalmeyer-Laughlin (KL) fractional quantum Hall ground state. Here, we construct generic families of chiral projected entangled pair states (chiral PEPS) with low bond dimension which, upon optimization, provide better variational energies than the KL Ansatz. The optimal PEPS exhibits chiral edge modes described by the Wess-Zumino-Witten SU(2)-level 1 model, as expected for the KL spin liquid. However, we find evidence that, in contrast to the KL state, the PEPS spin liquids have power-law dimer-dimer correlations and exhibit a gossamer long-range tail in the spin-spin correlations. We conjecture that these features are genuine to local chiral AFHM on bipartite lattices.