Dynnikov's connectedness conjecture for chaotic sections

Let $M_g^2$ be a surface in ${\mathbb T}^3$, let $B$ be a direction, and let ${\cal F}_B(M_g^2)$ denote the corresponding foliation or section system. Call ${\cal F}_B(M_g^2)$ chaotic when it has the chaotic behavior described in the surrounding discussion. Dynnikov's conjecture. If ${\cal F}_B(M_g^2)$ is chaotic, almost all the $B$-sections of $M_g^2$ consist in a single connected curve. The conjecture concerns the structure of chaotic level sections and is stated as a proposal from 2008; the source gives no resolution.