Strong $\mathbf g$-vector support conjecture

A cluster monomial is a monomial in cluster variables from one seed, its support is the set of variables with nonzero exponent, and its $\mathbf g$-vector is the product of the $\mathbf g$-vectors of its variables with multiplicities. Strong $\mathbf g$-vector support conjecture. Suppose two cluster monomials have the same $\mathbf g$-vector. If one is supported on a set ${\mathcal X}$ of cluster variables in a seed, and the other is supported on a set ${\mathcal X}'$ in another seed, then ${\mathcal X}={\mathcal X}'$, and the two seeds are related by a sequence of seed mutations that do not exchange variables in ${\mathcal X}$. The source says this is equivalent to the conjunction of the face-connectivity and $\mathbf g$-vector fan conjectures; no general resolution is given.