Residue reduction conjecture for generalized amplitudes

Let $m^{(k)}_n$ and $m^{(2)}_n$ denote the generalized amplitudes, and let $\text{base}_{k,n}$ and $\text{comb}_{k,n}$ be the collections of planar kinematic invariants defined by

$$\text{base}_{k,n}=\left\{\eta_{\lbrack 3,i\rbrack\cup \lbrack n-(k-i+1),n\rbrack}:i=3,\ldots,k\right\}$$

and

$$\text{comb}_{k,n}=\left\{\eta_{\lbrack j,j+r\rbrack\cup \lbrack n-r,n\rbrack}:j=3,\ldots,n-k,\ \text{and }r=1,\ldots,k-2\right\}.$$

Residue reduction conjecture. The $(k-2)(n-k-2)$-dimensional residue of $m^{(k)}_n$ on the subspace where $\eta_J=0$ for every $\eta_J\in\text{base}_{k,n}\cup\text{comb}_{k,n}$ can be identified with $m^{(2)}_n$; that is, there exists an identification of kinematics such that

$$\operatorname{Res}\left[m^{(k)}_n\right]_{\eta_J=0,\ \eta_J\in\text{base}_{k,n}\cup\text{comb}_{k,n}}=m^{(2)}_n.$$

This conjecture specifies a particular residue expected to reproduce the ordinary amplitude from the generalized one, extending the reduction suggested by the known examples. The source gives no resolution beyond presenting it as an all-$(k,n)$ conjecture.