Optimal Gromov capacity bound for Vianna's exotic Lagrangian tori

Let $T_{a,b,c}\subset \mathbb{C}P^2$ be Vianna's monotone Lagrangian torus associated with a Markov triple $(a,b,c)\in\mathfrak M$. A capacity conjecture. There is no monotone symplectic ball in $\mathbb{C}P^2\setminus T_{a,b,c}$, and

$$\inf_{(a,b,c)\in\mathfrak M}c_G(\mathbb{C}P^2;T_{a,b,c})=\frac{2\pi}{3},$$

where $2\pi/3$ is the capacity of the monotone ball. The preceding theorem gives the matching lower bound, while the conjecture asserts its optimality; for each individual torus the lower bound is strictly larger by some $\varepsilon_{a,b,c}>0$.