  I prove that the full Fock-Goncharov conjecture holds for
$\mathrm{Conf}_3^\times(\widetilde{\mathcal{F}\hspace{-1.6pt}\ell})$-- the
configuration space of triples of decorated flags in generic position. As a key
ingredient of this proof, I exhibit a maximal green sequence for the quiver of
the initial seed. I compute the Landau-Ginzburg potential $W$ on
$\mathrm{Conf}_3^\times(\widetilde{\mathcal{F}\hspace{-1.6pt}\ell})^\vee$
associated to the partial minimal model
$\mathrm{Conf}_3^\times(\widetilde{\mathcal{F}\hspace{-1.6pt}\ell}) \subset
\mathrm{Conf}_3(\widetilde{\mathcal{F}\hspace{-1.6pt}\ell})$. The integral
points of the associated "cone" ${\Xi:=\left\{W^T \geq 0\right\} \subset
\mathrm{Conf}_3^\times(\widetilde{\mathcal{F}\hspace{-1.6pt}\ell})^\vee\left({\mathbb{R}^T}\right)}$
parametrize a basis for
$\mathcal{O}\left(\mathrm{Conf}_3(\widetilde{\mathcal{F}\hspace{-1.6pt}\ell})\right)
= \bigoplus \left(V_\alpha \otimes V_\beta \otimes V_\gamma \right)^G$ and
encode the Littlewood-Richardson coefficients $c^\gamma_{\alpha \beta}$. In the
initial seed, the inequalities defining $\Xi$ are exactly Zelevinsky's tail
positivity conditions. I exhibit a unimodular $p^*$ map that identifies $W$
with the potential of Goncharov-Shen on
$\mathrm{Conf}_3^\times(\widetilde{\mathcal{F}\hspace{-1.6pt}\ell})$ and $\Xi$
with the Knutson-Tao hive cone.
