The simply connected level-set conjecture for checkerboard switching

Let $R\in\mathbb{N}^p$, $C\in\mathbb{N}^q$, and let $\mathbf{M}\in\mathcal{M}_{p,q}(\{-1,0,1\})$. For each integer $i$, let $\mathcal{P}_i(\mathbf{M})$ denote the set of cells at level $i$ associated with $\mathbf{M}$, and let $\mathbf{C}_{i,i+1,k,k+1}$ denote the corresponding elementary $2\times2$ checkerboard move. Write $\mathbf{A}\rightarrow\mathbf{A}'$ for the switching relation on matrices in $\mathcal{M}(R,C)$. The simply connected level-set conjecture. We have $\mathbf{A}\rightarrow\mathbf{A}'$ for all $\mathbf{A},\mathbf{A}'\in\mathcal{M}(R,C)$ such that $\mathbf{A}'-\mathbf{A}=\mathbf{M}$ if and only if both of the following conditions hold: (i) there exists $\mathbf{T}\in\mathcal{M}_{p-1,q-1}(\mathbb{N})$ such that

$$\mathbf{M}=\sum_{i,k}t_{ik}\mathbf{C}_{i,i+1,k,k+1};$$

(ii) for every $i$, each connected component of $\mathcal{P}_i(\mathbf{M})$ is simply connected. The conjecture removes condition (iii) from the preceding sufficient theorem and predicts that failure of simple connectivity is exactly the obstruction to universal switchability.