The necessity conjecture for intermediate crossing–uncrossing pairs in Go-diagrams

Let $D'$ be a Go-diagram in a partition $\lambda$, and let $c\prec c'$ be a crossing and uncrossing pair in $D'$ with $D'(c)=\circ$. Let $D$ be obtained by replacing the stones at $c$ and $c'$ with $+$. Let $i<j$ be the labels of the pipes crossing at $c$ in $D'$. The necessity conjecture. Suppose that, after recolouring stones appropriately, $D$ is a Go-diagram with no cells in the forbidden configuration, and that there is a label $k$ with $i<k<j$ such that $k$ forms a crossing and uncrossing pair with $i$ or $j$ between $c$ and $c'$ in $D'$. Then

$$\mathcal{D}_{D'}\not\subset\overline{\mathcal{D}_D}.$$

The source presents this as the conjectured necessity of the second condition in the preceding closure conjecture. It is motivated by the failure of the first condition alone, but no proof or resolution is supplied.