Sodin's conjecture on asymptotic dependence of local lengths

Let $x\in \mathcal{S}^{2}$ and, for $t\in \mathbb{R}$, let $\mathcal{L}_{l;x}^{t}$ denote the random length of the unique component of $f_l^{-1}(t)$ that contains $x$ inside, or $0$ if $f_l$ does not cross the level $t$. For random variables $X$ and $Y$, write $\operatorname{Corr}(X,Y)=\operatorname{Cov}(X,Y)/(\sqrt{\operatorname{Var}(X)}\sqrt{\operatorname{Var}(Y)})$. Sodin's conjecture. For every $x\in \mathcal{S}^{2}$ and $t_{1},t_{2}\in \mathbb{R}$, the local lengths are asymptotically fully dependent:

$$\operatorname{Corr}\left(\mathcal{L}_{l;x}^{t_{1}},\mathcal{L}_{l;x}^{t_{2}}\right)=1-o_{l\rightarrow\infty}(1).$$

This conjecture would explain the asymptotic full dependence observed for global level lengths and the resulting degeneracy of the defect variance. The surrounding discussion presents it as a possible explanation and does not provide a proof or resolution.