PL order on the RC cylinder and the BC order on the TC cylinder. On a finite-size system, different states being stabilized on different geometries suggest energetically degenerate competing states [@gong2015honeycomb]. This is corroborated by a comparison of the bulk energies of the two states on different system sizes [@supplemental]. On the large cylinders, we find the energies of the two states to be quite close to each other. For the spin-$1$ model, the energy difference between the RC ($L_y = 8$) and TC ($L_y = 6$) cylinders is only $0.2\%$. Thus, our DMRG results strongly suggest the (quasi-) degeneracy of the BC and PL states in the $J$-$K$ model, even after the quantum fluctuations effects are considered, which is also consistent with the flavor-wave theory results to be presented below.

    \
    

*Stiffness of the BC/PL orders in the $J$-$K$ model.*— To address the mechanisms for breaking the BC/PL degeneracy in the BC and PL phases of the $J$-$K$ model, we perform the flavor-wave theory analysis [@Bauer2012; @Lai_SU(3)_gapless], which considers partial quantum fluctuations going beyond the semi-classical picture. Within the flavor wave calculations for BC and PL, we consider a square lattice consisting of $16$ sublattice per unit cell illustrated 