Exotic monotone torus conjecture for $\mathbb{CP}^1\times\mathbb{CP}^1$

Let $T^c_{r,0}$ be a Lagrangian torus in $\mathbb{CP}^1\times\mathbb{CP}^1$ of type $T(1,2,9)$, and let a family of Maslov-index-$2$ holomorphic discs mean a family of such discs with boundary on the torus. A monotone Lagrangian torus is one for which the symplectic area is proportional to the Maslov index on relative homotopy classes. Exotic monotone torus conjecture. There is a monotone $T(1,2,9)$ torus, of the form $T^c_{r,0}$, in $\mathbb{CP}^1\times\mathbb{CP}^1$, bounding $9$ families of Maslov-index-$2$ holomorphic discs, that is not symplectomorphic to the monotone Chekanov torus nor to the monotone Clifford torus. This predicts an exotic monotone Lagrangian torus in $\mathbb{CP}^1\times\mathbb{CP}^1$ distinct from the known Clifford and Chekanov examples; the supplied text gives no resolution status.