Nonsolvability conjecture for the three-column restricted maze class

Let $\mathcal{M}$ be the class of all mazes, and let $\mathcal{N}_3\subset\mathcal{M}$ be the set of mazes for which the only horizontal edges removed between the pairs of columns $(c_{-4},c_{-3})$ and $(c_3,c_4)$ are HNEs. Three-column restricted nonsolvability conjecture. There is no algorithm that solves $\mathcal{N}_3$. The paper presents this as a stronger proposed negative result and does not resolve it.