Augmentation-polynomial conjecture for monotone Lagrangian two-tori

Let $L\subset(\mathbb{C}P^2,\omega_{\operatorname{FS}})$ be an embedded monotone Lagrangian two-torus, and let $\Lambda$ be the Legendrian lift of its canonical threefold Bohr–Sommerfeld cover. Assume that

$$H_0(\mathcal{A}(\Lambda),\partial)=\mathbb{C}[\mu^{\pm1},\lambda^{\pm1}]/\langle \mathcal{A}ug_{\Lambda}(\mu,\lambda)\rangle,$$

where $\mathcal{A}ug_{\Lambda}(\mu,\lambda)$ is the augmentation polynomial, uniquely determined up to multiplication by a unit under these assumptions. Let $\mathfrak{P}_L(u,v)$ denote the superpotential of $L$. Augmentation-polynomial conjecture. Up to multiplication by a unit, the superpotential can be recovered from the augmentation polynomial by

$$\mathfrak{P}_{L}(u,v)=\mathcal{A}ug_{\Lambda}(u^3,v)\in u\cdot\mathbb{C}[u^{\pm 3},v^{\pm1}],$$

of ideals, using suitable choices of bases of $H_1(L)$ and $H_1(\Lambda)$, capping paths, and an appropriate normalization of $\mathcal{A}ug_{\Lambda}(\mu,\lambda)$. This proposes a direct relation between the augmentation variety of the Legendrian lift and the Maslov-two disc count defining the Lagrangian superpotential; the source confirms the relation for the monotone Clifford and Chekanov tori by hands-on comparison but states the general assertion as a conjecture.