Vanishing Weinstein width conjecture for Vianna's exotic Lagrangian tori

For each Markov triple $(a,b,c)\in\mathfrak M$, let $T_{a,b,c}\subset\mathbb{C}P^2$ be Vianna's monotone Lagrangian torus, and let ${\frak w}_{\text{DW}}(T_{a,b,c};\mathbb{C}P^2)$ denote its Weinstein width, defined using Darboux--Weinstein charts and a fixed Riemannian metric. An Weinstein-width conjecture.

$$\inf_{(a,b,c)\in\mathfrak M}{\frak w}_{\text{DW}}(T_{a,b,c};\mathbb{C}P^2)=0.$$

The conjecture is proposed as a way to examine the conjectural ergodic behavior of the family $T_{a,b,c}$; the source establishes only that each individual Weinstein width is positive.