Refined generic level-curve contact conjecture

Let $\phi$ satisfy condition (A1). For $\lambda\in\mathbb{T}$, let $\mathcal{L}_{\lambda}(\phi)$ have $L_{\lambda}$ components near $(1,1)$, and let $\mathcal{Z}_{\tilde{p}}$ have $L_0$ branches there. Write the corresponding components as $z_1=\psi^\lambda_\ell(z_2)$ and the components for the exceptional value as $z_1=\psi^0_\ell(z_2)$. Refined generic level-curve contact conjecture. For almost every pair $\lambda,\mu\in\mathbb{T}$,

$$L_{\lambda}=L_0=L_{\mu}.$$

After reordering the components of $\mathcal{L}_{\lambda}(\phi)$ and $\mathcal{L}_{\mu}(\phi)$ near $(1,1)$, the contact order of $z_1=\psi^0_\ell(z_2)$ at $(1,1)$ equals the contact order between $z_1=\psi^\mu_\ell(z_2)$ and $z_1=\psi^\lambda_\ell(z_2)$ at $(1,1)$ for $1\leq\ell\leq L_0$. The preceding argument establishes the analogous lower-bound statement, while this equality and the exact component count remain conjectural.