Fei's vertex-coefficient and saturated-support conjecture

Let ${\mathcal A}$ be a TSSS cluster algebra with principal coefficients. For a coefficient-free cluster monomial, let its Newton polytope be the convex polytope determined by its Laurent expansion, and let $co_p$ denote the coefficient associated with a point $p$; let the support of an $F$-polynomial be the set of exponent vectors having nonzero coefficient, and call this support saturated when it contains every lattice point of its convex hull. Fei's conjecture. (i) A point $p$ in the Newton polytope associated to any coefficient-free cluster monomial is a vertex if and only if $co_p=1$. (ii) The support of the $F$-polynomial of any cluster monomial is saturated. These statements are presented as conjectures attributed to Jiarui Fei; the supplied text gives no resolution status.