split the degeneracies, but they lower the energies of BC/PL more than those of $(\pi/2,\pi/2)$ AFM$_{a/b}^*$. The $4$-site ring exchanges split the double degeneracy in both BC/PL and $(\pi/2,\pi/2)$ AFM$^*_{a/b}$. The competing phases become BC and $(\pi/2,\pi/2)$ AFM$_a^*$. The boundary can be determined by the equation, $K_1 + 2K_2 = 4 \mathfrak{R}_3 \Rightarrow K = 2K_2 - 4 \mathfrak{R}_3.$

Flavor wave theory for $(\pi/2,\pi/2)$ bicollinear order and $(\pi/2,\pi/2)$ plaquette order {#App:FLWT}
============================================================================================

Within the flavor wave calculation for BC and PL with $16$ sublattice per unit cell illustrated in Fig. 2(b) in the main text, we associate $3$ Schwinger-bosons at each site $i$, $b_{i\alpha=x,y,z}$, to the states under SU(3) time-reversal invariant basis, $|x\ra,~|y\ra,~|z\ra$, where $b^\dagger_{i\alpha} |vac\ra = |\alpha\ra$ with $|vac\ra$ being the vacuum state of the Schwinger bosons. The bosons satisfy a local constraint $$\begin{aligned}
\label{Eq:constraint}
\sum_{\alpha} b^\dagger_{i \alpha}b_{i \alpha} = 1.\end{aligned}$$ The model Hamiltonian in terms of spin operators can be re-expressed in terms of the bosons, $$\begin{aligned}
H = \sum_{i, \delta_n} \left[ J_n {\bf S}_i \cdot {\bf S}_j + K_n \left( {\bf S}_i \cdot {\bf S}_j \right)^2 \right] = \sum_{i, \delta_n, \alpha, \beta} \left[ J_nb^\dagger_{i \alpha} b_{i \alpha} b^\dagger_{j\beta} b_{i\beta} + \left(K_n - J_n\right) b_{i\alpha}^\dagger b^\dagger_{j \alpha} b_{i\beta} 