Novikov's stability-zone area and topology conjecture

Let $M_g^2\subset\mathbb{T}^3$ be a generic embedded surface, let $\ell$ be a stability-zone label, and let $\mathcal{D}_{\ell}(M_g^2)$ denote its stability region. Write $\|\ell\|$ for the norm of the label and let $f:\mathbb{T}^3\to\mathbb{R}$ be a generic smooth function, with $\mathcal{D}_{\ell}(f)$ its corresponding stability zone. Novikov's conjecture. The area of a stability region $\mathcal{D}_{\ell}(M_g^2)$ does not exceed $C/\|\ell\|^3$ for some constant $C$ depending only on $M_g^2$, and the sets $\mathcal{D}_{\ell}(f)$ are connected and simply connected. The claim concerns quantitative shrinking and global topology of stability zones; the source presents it as not well understood and gives no resolution.