{\bf d}_1 \otimes \cdots \otimes \bar{\bf d}_{n-1} \right] = (\bar{\bf d}_1 \cdot {\bf d}_n )(\bar{\bf d}_2 \cdot {\bf d}_1) \cdots (\bar{\bf d}_n \cdot {\bf d}_{n-1}) .\end{aligned}$$ The cases of $n=3,4$ correspond to those shown in the main text.

Site-factorized wavefunction energies of competing orders {#App:SFW}
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In the present work, within the site-factorized wavefunction studies we find four competing phases, $(\pi/2,\pi/2)$ bicollinear order(BC), $(\pi/2,\pi/2)$ plaquette order(PL), $(\pi/2,\pi/2)$ AFM$_a^*$, and $(\pi/2,\pi/2)$ AFM$_b^*$, in the $J$-$K$-$\mathfrak{R}$ model as $$\begin{aligned}
&& \mathcal{E}_{BC}= K_1  + K_2 + 2(K_3 - J_3)- 2\mathfrak{R}_3,\\
&& \mathcal{E}_{PL}= K_1  + K_2 + 2(K_3 - J_3) - 2\mathfrak{R}_3+ \frac{1}{2}\mathfrak{R}_4,\\
&& \mathcal{E}_{(\pi/2, \pi/2)AFM_a^*} = \frac{3}{4}K_1 + \frac{1}{2} K_2 + 2(K_3 - J_3) -\mathfrak{R}_3, \\
 && \mathcal{E}_{(\pi/2, \pi/2)AFM_b^*} = \frac{3}{4}K_1 + \frac{1}{2} K_2 + 2(K_3 - J_3) -\mathfrak{R}_3+ \frac{1}{4}\mathfrak{R}_4.\end{aligned}$$ In the absence of the ring exchanges, $\mathfrak{R}_{3/4}$, BC and PL are degenerate and so are $(\pi/2,\pi/2)$ AFM$_a^*$ and $(\pi/2,\pi/2)$ AFM$^*_b$. The boundary between degenerate BC/PL and degenerate $(\pi/2,\pi/2)$ AFM$_{a/b}^*$ in the absence of ring exchange terms is determined by $K_1 = -2 K_2 \Rightarrow K = 2 K_2$, where we explicitly use the definition $K_1  = -K$. We remark that the site-factorized wavefunction analysis in the present spin-$1$ model suggests that in 