being $\sim 0.001\%$.) This implies that the previous conclusion suggesting that $1/S$ quantum fluctuations stabilize the PL phase in an extended parameter range cannot apply to the $J_1$-$J_2$-$J_3$ model [@Chubukove_PL2012]. Going “slightly” beyond the $J$-only model by considering the effects of $K_1 = K_3 = -K$ with $K_2=0$, we find that increasing $K$ significantly enhances the threshold values of $J_1$ for stable BC and PL orders. Taking $K=0.1, K_2 =0$, we find threshold values $J^{BC}_1 \simeq 0.42$ for BC and $J^{PL}_1 \simeq 0.5$ for PL (see Supplemental Material [@supplemental]). On the other hand, increasing $K_2 > 0$ reduces the stable BC and PL regimes.

*Ring exchanges stabilizing BC.*— We are now in position to address the role of the ring exchange interactions in the wide parameter regime where the BC and PL phases are degenerate, with both having positive stiffness. In the presence of the ring exchanges, the energy corrections, within the site-factorized wavefunction, to each order are $$\begin{aligned}
 &\mathcal{E}^{BC}_\mathfrak{R} =  -2 \mathfrak{R}_3,~~~&\mathcal{E}^{PL}_\mathfrak{R} = -2 \mathfrak{R}_3 + \frac{\mathfrak{R}_4}{2},\\
&\mathcal{E}^{AFM^{*}_a}_\mathfrak{R} = - \mathfrak{R}_3, ~~~& \mathcal{E}^{AFM^{*}_b}_\mathfrak{R} = - \mathfrak{R}_3 + \frac{\mathfrak{R}_4}{4}.\end{aligned}$$ The $3$-site ring exchange interaction, $\mathfrak{R}_3$, does not split the degeneracy of BC/PL and that of AFM$^*_{a/b}$, although it does lower 