Fiberwise isotopy conjecture for tropical Lagrangian lifts

Let $V$ be a tropical curve and let $L_V$ be a homologically minimal Lagrangian lift. Let $\operatorname{Def}_{\mathcal L}$ denote the sheaf governing the relevant tropical line-bundle deformations. Let $\operatorname{Aff}^*_V(U)$ be the sheaf of invertible locally integral affine functions from $U$ to $\mathbb R$, and let $H^1(V,\operatorname{Aff}^*_V)_0$ be the connected component containing the identity.

Fiberwise isotopy conjecture. The subspace of $H^1(L_V,\mathbb R)$ arising from flux classes of fiberwise Lagrangian isotopies is identified with $H^0(V,\operatorname{Def}_{\mathcal L})$. Moreover,

$$\{\text{fiberwise isotopies}\}/\{\text{fiberwise Hamiltonian isotopies}\}\simeq H^1(V,\operatorname{Aff}^*_V)_0.$$

This predicts that fiberwise symplectic deformations of a tropical Lagrangian correspond to tropical line-bundle data on the mirror curve. The source relates the latter group to the tropical Picard group, but does not prove the full identification.