Width-one conjecture for rational toric hypersurfaces

Let $P$ be a $d$-dimensional lattice polytope. Suppose that, for every field $K$, every irreducible toric hypersurface $Z\subset\mathbb{T}^d_K$ with Newton polytope $P$ is birational over $K$ to $\mathbb{A}^{d-1}_K$. A lattice polytope has width one if it admits a lattice projection onto the segment $[0,1]$.

Width-one conjecture. Under the stated hypothesis, $P$ has width one.

The preceding lemma establishes that width one implies rationality for non-degenerate toric hypersurfaces. The conjecture asserts a converse under the stronger universal rationality assumption; the supplied text gives no resolution status.