Three-ary factorization conjecture for generalized biadjoint amplitudes

Fix an ordered set partition $(I,J,K)$ of ${1,\ldots,n}$, where

$$I=\{i,i+1,\ldots,j-1\},\qquad J=\{j,j+1,\ldots,k-1\},\qquad K=\{k,k+1,\ldots,i-1\}.$$

Let $\eta_{(I_1,JK_2)}$, $\eta_{(IJ_2,K_1)}$, and $\eta_{(KI_2,J_1)}$ be the three residue parameters used in the statement, and let $m^{(3)}(Ibc)$, $m^{(3)}(Jac)$, and $m^{(3)}(Kab)$ denote the corresponding lower-point amplitudes.

Three-ary factorization conjecture. To leading order in these three parameters,

$$m^{(3)}_n=\left(\frac{1}{\eta_{(I_1,J_1,K_1)}}+\frac{1}{\eta_{(I_1,J_1,K_1)}}\right)\!\cdot\!\left(\frac{m^{(3)}(Ibc)m^{(3)}(Jac)m^{(3)}(Kab)}{\eta_{(IJ_2,K_1)}\eta_{(KI_2,J_1)}\eta_{(I_1,JK_2)}}\right),$$

where

$$\eta_{(I_1,J_1,K_1)}+\eta_{(I_1,K_1,J_1)}=\eta_{(I_1,JK_2)}+\eta_{(IJ_2,K_1)}+\eta_{(KI_2,J_1)}.$$

This is an explicit proposed factorization formula for the $k=3$ case; the notation and the duplicated denominator term should be checked against the source.