Hickingbotham's conjecture on low-connectivity ghost-edges

Let $k\in\mathbb{N}$, let $G$ be a connected graph with $tw(G)\leq k$, and let $xy\in E(G^c)$. A $k$-ghost-edge is a nonedge $xy$ such that every tree decomposition $(T,\mathcal{B})$ of $G$ with width at most $k$ has a bag containing both $x$ and $y$. Hickingbotham's conjecture. If there are at most $k$ internally vertex-disjoint $(x,y)$-paths in $G$, then $xy$ is not a $k$-ghost-edge of $G$. The paper's abstract states that this conjecture is wrong, so the claim is refuted.