The path-graph invariant-subspace completeness conjecture

Let $G=P_n$ be the path graph on $n$ cells, and let $\mathcal D_{G,\ell}$ denote the collection of transformations whose invariant subspaces are being classified by the algorithm in the preceding example. Path-graph completeness conjecture. The algorithm in the previous example gives all of the $\mathcal D_{G,\ell}$-invariant subspaces for the path $G=P_n$. This conjecture asserts completeness of the proposed classification algorithm for invariant subspaces of path networks; the source provides no resolution.