Martin's scaffold duality conjecture for association schemes

A standard planar scaffold is a weakly connected planar scaffold whose edge weights are among the adjacency matrices $A_0,\dots,A_c$ of an association scheme. For sequences of such scaffolds, write $\mathsf{S}^{\dagger}$ for the dual scaffold, let $|X|$ denote the number of vertices of the association scheme, and let $n_{j,i}$ and $n_i'$ denote the numbers of nodes of $\mathsf{S}_{j,i}$ and $\mathsf{S}_i'$, respectively. Let

$$\{\mathsf{S}_{j,i}\}_{i=1}^{k_j}\quad (1\leqslant j\leqslant h),\qquad \{\mathsf{S}_i'\}_{i=1}^{k'}$$

be sequences whose scaffolds in each sequence have the same order, and let $=_1,\dots,=_h,='\,\in\{=,\ne\}$ and $a_{j,i},b_i\in\mathbb{C}$. Martin's scaffold duality conjecture. If, for every association scheme with $d\geqslant c$ classes,

$$\sum_{i=1}^{k_j} a_{j,i}\mathsf{S}_{j,i} =_j0\quad (1\leqslant j\leqslant h)\quad\Longrightarrow\quad\sum_{i=1}^{k'}b_i\mathsf{S}_i'='0,$$

then, for every association scheme with $d\geqslant c$ classes,

$$\sum_{i=1}^{k_j}a_{j,i}|X|^{n_{j,i}}\mathsf{S}_{j,i}^{\dagger} =_j0\quad (1\leqslant j\leqslant h)\quad\Longrightarrow\quad\sum_{i=1}^{k'}b_i|X|^{n_i'}\mathsf{S}_i^{\prime\dagger}='0.$$

This is a strengthened and formalized version of Martin's proposed duality principle: universal implications between equations of scaffolds weighted by adjacency matrices should remain valid after dualizing the scaffolds and interchanging adjacency-matrix and primitive-idempotent weights. Its status is not resolved in the supplied source.