The extension conjecture for non-toric staged tree models

Let $\mathcal T$ be a staged tree with non-toric $V(\ker \varphi_{\mathcal T})$ that uses color set $S$. Let $\mathcal T'$ be a staged tree such that its restriction to color set $S$ gives $\mathcal T$. Extension conjecture. Then $\ker \varphi_{\mathcal T'}$ is not toric under any linear change of variables. This conjecture proposes that non-toric structure persists when a staged tree is extended while retaining the original color set. The paper gives no proof or resolution, so the conjecture remains open.