Backward Orbit Conjecture for Latt\'es Maps

For a Latt`es map $\varphi :{\mathbb{P}}^1\to {\mathbb{P}}^1$ defined over a number field $K$, we prove a conjecture on the integrality of points in the backward orbit of $P\in {\mathbb{P}}^1(\bar{K})$ under $\varphi$.