Sign-coherence and constant-term conjectures for cluster-algebra matrices and F-polynomials

Let $C'$ be the $C$-matrix of a seed in a cluster algebra, and call its columns \mathbf{c}-vectors. Let $F'_i(y)$ be the associated $F$-polynomials.

Sign-coherence and constant-term conjectures. (i) Every column of $C'$ is a nonzero vector, and its nonzero components are either all positive or all negative. (ii) Every $F$-polynomial has constant term $1$.

These are presented as two fundamental conjectures on cluster algebras. The supplied text gives no resolution status.