Dynnikov's stability-zone size and topology conjecture

Let $M_g^2\subset{\mathbb T}^3$ be a generic embedded surface, let $\ell$ be a soul label, and let ${\cal D}_{\ell}(M_g^2)$ denote its stability region; write $\|\ell\|$ for the norm of the label. For a function $f$ on ${\mathbb T}^3$, let ${\cal D}_{\ell}(f)$ denote the corresponding stability zone. Dynnikov's conjecture. The area of a stability region ${\cal D}_{\ell}(M_g^2)$ does not exceed $C/\|\ell\|^3$ for some constant $C$ that depends only on $M_g^2$. The sets ${\cal D}_{\ell}(f)$ are connected and simply connected. These properties concern quantitative decay and topology of stability zones; the source lists them among properties that were not well understood and gives no resolution.