The constant-term-one conjecture for F-polynomials

Fix an initial exchange matrix $B^0$, an initial vertex $t_0$, and a vertex $t$. Let $F_{\ell;t}^{B^0;t_0}(y_1,\dots,y_n)$ be the associated $F$-polynomial of the cluster variable indexed by $\ell$. Constant-term-one conjecture. Every polynomial $F_{\ell;t}^{B^0;t_0}(y_1,\dots,y_n)$ has constant term $1$. This strengthens the proved fact that no $y_j$ divides an $F$-polynomial. The paper notes equivalent extremal-monomial formulations and proves the conjecture in additional special cases, but leaves it open in general.