Singleton end-image conjecture for Tutte embeddings

Let $\mathcal M_k$ be the supercritical random planar map and let $H_k$ be its Tutte embedding into $\overline{\mathbb D}$. For an end $\omega$ of $\mathcal M_k$, define its image $H_k(\omega)\subset\overline{\mathbb D}$ as in the source. Singleton end-image conjecture. Almost surely, $H_k(\omega)$ is a singleton for every end $\omega$ of $\mathcal M_k$. This is motivated by the fact that singular points of the limiting LQG metric are totally disconnected and are expected to correspond to ends of the map; the conjecture asserts that each discrete end has a unique Euclidean image under the embedding.