General factorization conjecture for cyclic block partitions

Fix an ordered set partition $\mathbf{S}=(S_1,\ldots,S_k)$ of $\{1,\ldots,n\}$ into $k$ cyclically contiguous interval blocks. Let $d$ be the number of blocks with size at least $2$, and let $\mathcal{N}_{\mathbf{S}}$ denote the associated set of distinct nonzero kinematic blades.

Block-partition factorization conjecture. There is an ordering $(\mathbf{T}_1,\ldots,\mathbf{T}_{(d-1)(k-1)-1},\mathbf{S})$ with $\{\mathbf{T}_1,\ldots,\mathbf{T}_{(d-1)(k-1)-1}\}=\mathcal{N}_{\mathbf{S}}$ such that the iterated residue is nonzero and

$$\operatorname{Res}_{\eta_{(\mathbf{T}_{(d-1)(k-1)-1})}=0}\!\left(\cdots\operatorname{Res}_{\eta_{(\mathbf{T}_1)}=0}\!\left(\operatorname{Res}_{\eta_{(\mathbf{S})}=0}(m^{(k)}_n)\right)\cdots\right)=\prod_{\ell=1}^{d}m^{(k)}_{n_\ell},$$

under a suitable identification of kinematic parameters, where $n_1+\cdots+n_d=n+d(k-1)$.

This extends the explicit factorization proposal to arbitrary $k$ and block degeneracies. The supplied text gives no resolution, so the conjecture remains open.