Fei's saturation conjecture for principal-coefficient cluster variable Newton polytopes

Let $\mathcal{A}(\Sigma)$ be a cluster algebra with principal coefficients at the seed $\Sigma$. A Newton polytope is the convex hull of the exponent vectors of the monomials occurring in a Laurent polynomial, and it is saturated when every lattice point in the polytope occurs as such an exponent vector. Fei's saturation conjecture. The Newton polytopes of cluster variables, written as Laurent polynomials in $\Sigma$, are saturated. This conjecture was proved in the paper for cluster algebras of types $A$ and $D$, while the general case remains open.